Calculus Examples

$\int x^{\frac{1}{3}} {(x^{\frac{4}{3}} + 5)}^{8} d x$

Step 1

This integral could not be completed using u-substitution. Mathway will use another method.

Step 2

Simplify.

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Step 2.1

Use the Binomial Theorem.

$\int x^{\frac{1}{3}} ({(x^{\frac{4}{3}})}^{8} + 8 {(x^{\frac{4}{3}})}^{7} \cdot 5 + 28 {(x^{\frac{4}{3}})}^{6} \cdot 5^{2} + 56 {(x^{\frac{4}{3}})}^{5} \cdot 5^{3} + 70 {(x^{\frac{4}{3}})}^{4} \cdot 5^{4} + 56 {(x^{\frac{4}{3}})}^{3} \cdot 5^{5} + 28 {(x^{\frac{4}{3}})}^{2} \cdot 5^{6} + 8 x^{\frac{4}{3}} \cdot 5^{7} + 5^{8}) d x$

Step 2.2

Simplify each term.

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Step 2.2.1

Multiply the exponents in ${(x^{\frac{4}{3}})}^{8}$ .

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Step 2.2.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\int x^{\frac{1}{3}} (x^{\frac{4}{3} \cdot 8} + 8 {(x^{\frac{4}{3}})}^{7} \cdot 5 + 28 {(x^{\frac{4}{3}})}^{6} \cdot 5^{2} + 56 {(x^{\frac{4}{3}})}^{5} \cdot 5^{3} + 70 {(x^{\frac{4}{3}})}^{4} \cdot 5^{4} + 56 {(x^{\frac{4}{3}})}^{3} \cdot 5^{5} + 28 {(x^{\frac{4}{3}})}^{2} \cdot 5^{6} + 8 x^{\frac{4}{3}} \cdot 5^{7} + 5^{8}) d x$

Step 2.2.1.2

Multiply $\frac{4}{3} \cdot 8$ .

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Step 2.2.1.2.1

Combine $\frac{4}{3}$ and $8$ .

$\int x^{\frac{1}{3}} (x^{\frac{4 \cdot 8}{3}} + 8 {(x^{\frac{4}{3}})}^{7} \cdot 5 + 28 {(x^{\frac{4}{3}})}^{6} \cdot 5^{2} + 56 {(x^{\frac{4}{3}})}^{5} \cdot 5^{3} + 70 {(x^{\frac{4}{3}})}^{4} \cdot 5^{4} + 56 {(x^{\frac{4}{3}})}^{3} \cdot 5^{5} + 28 {(x^{\frac{4}{3}})}^{2} \cdot 5^{6} + 8 x^{\frac{4}{3}} \cdot 5^{7} + 5^{8}) d x$

Step 2.2.1.2.2

Multiply $4$ by $8$ .

$\int x^{\frac{1}{3}} (x^{\frac{32}{3}} + 8 {(x^{\frac{4}{3}})}^{7} \cdot 5 + 28 {(x^{\frac{4}{3}})}^{6} \cdot 5^{2} + 56 {(x^{\frac{4}{3}})}^{5} \cdot 5^{3} + 70 {(x^{\frac{4}{3}})}^{4} \cdot 5^{4} + 56 {(x^{\frac{4}{3}})}^{3} \cdot 5^{5} + 28 {(x^{\frac{4}{3}})}^{2} \cdot 5^{6} + 8 x^{\frac{4}{3}} \cdot 5^{7} + 5^{8}) d x$

Step 2.2.2

Multiply the exponents in ${(x^{\frac{4}{3}})}^{7}$ .

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Step 2.2.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\int x^{\frac{1}{3}} (x^{\frac{32}{3}} + 8 x^{\frac{4}{3} \cdot 7} \cdot 5 + 28 {(x^{\frac{4}{3}})}^{6} \cdot 5^{2} + 56 {(x^{\frac{4}{3}})}^{5} \cdot 5^{3} + 70 {(x^{\frac{4}{3}})}^{4} \cdot 5^{4} + 56 {(x^{\frac{4}{3}})}^{3} \cdot 5^{5} + 28 {(x^{\frac{4}{3}})}^{2} \cdot 5^{6} + 8 x^{\frac{4}{3}} \cdot 5^{7} + 5^{8}) d x$

Step 2.2.2.2

Multiply $\frac{4}{3} \cdot 7$ .

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Step 2.2.2.2.1

Combine $\frac{4}{3}$ and $7$ .

$\int x^{\frac{1}{3}} (x^{\frac{32}{3}} + 8 x^{\frac{4 \cdot 7}{3}} \cdot 5 + 28 {(x^{\frac{4}{3}})}^{6} \cdot 5^{2} + 56 {(x^{\frac{4}{3}})}^{5} \cdot 5^{3} + 70 {(x^{\frac{4}{3}})}^{4} \cdot 5^{4} + 56 {(x^{\frac{4}{3}})}^{3} \cdot 5^{5} + 28 {(x^{\frac{4}{3}})}^{2} \cdot 5^{6} + 8 x^{\frac{4}{3}} \cdot 5^{7} + 5^{8}) d x$

Step 2.2.2.2.2

Multiply $4$ by $7$ .

$\int x^{\frac{1}{3}} (x^{\frac{32}{3}} + 8 x^{\frac{28}{3}} \cdot 5 + 28 {(x^{\frac{4}{3}})}^{6} \cdot 5^{2} + 56 {(x^{\frac{4}{3}})}^{5} \cdot 5^{3} + 70 {(x^{\frac{4}{3}})}^{4} \cdot 5^{4} + 56 {(x^{\frac{4}{3}})}^{3} \cdot 5^{5} + 28 {(x^{\frac{4}{3}})}^{2} \cdot 5^{6} + 8 x^{\frac{4}{3}} \cdot 5^{7} + 5^{8}) d x$

Step 2.2.3

Multiply $5$ by $8$ .

$\int x^{\frac{1}{3}} (x^{\frac{32}{3}} + 40 x^{\frac{28}{3}} + 28 {(x^{\frac{4}{3}})}^{6} \cdot 5^{2} + 56 {(x^{\frac{4}{3}})}^{5} \cdot 5^{3} + 70 {(x^{\frac{4}{3}})}^{4} \cdot 5^{4} + 56 {(x^{\frac{4}{3}})}^{3} \cdot 5^{5} + 28 {(x^{\frac{4}{3}})}^{2} \cdot 5^{6} + 8 x^{\frac{4}{3}} \cdot 5^{7} + 5^{8}) d x$

Step 2.2.4

Multiply the exponents in ${(x^{\frac{4}{3}})}^{6}$ .

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Step 2.2.4.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\int x^{\frac{1}{3}} (x^{\frac{32}{3}} + 40 x^{\frac{28}{3}} + 28 x^{\frac{4}{3} \cdot 6} \cdot 5^{2} + 56 {(x^{\frac{4}{3}})}^{5} \cdot 5^{3} + 70 {(x^{\frac{4}{3}})}^{4} \cdot 5^{4} + 56 {(x^{\frac{4}{3}})}^{3} \cdot 5^{5} + 28 {(x^{\frac{4}{3}})}^{2} \cdot 5^{6} + 8 x^{\frac{4}{3}} \cdot 5^{7} + 5^{8}) d x$

Step 2.2.4.2

Cancel the common factor of $3$ .

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Step 2.2.4.2.1

Factor $3$ out of $6$ .

$\int x^{\frac{1}{3}} (x^{\frac{32}{3}} + 40 x^{\frac{28}{3}} + 28 x^{\frac{4}{3} \cdot (3 (2))} \cdot 5^{2} + 56 {(x^{\frac{4}{3}})}^{5} \cdot 5^{3} + 70 {(x^{\frac{4}{3}})}^{4} \cdot 5^{4} + 56 {(x^{\frac{4}{3}})}^{3} \cdot 5^{5} + 28 {(x^{\frac{4}{3}})}^{2} \cdot 5^{6} + 8 x^{\frac{4}{3}} \cdot 5^{7} + 5^{8}) d x$

Step 2.2.4.2.2

Cancel the common factor.

$\int x^{\frac{1}{3}} (x^{\frac{32}{3}} + 40 x^{\frac{28}{3}} + 28 x^{\frac{4}{3} \cdot (3 \cdot 2)} \cdot 5^{2} + 56 {(x^{\frac{4}{3}})}^{5} \cdot 5^{3} + 70 {(x^{\frac{4}{3}})}^{4} \cdot 5^{4} + 56 {(x^{\frac{4}{3}})}^{3} \cdot 5^{5} + 28 {(x^{\frac{4}{3}})}^{2} \cdot 5^{6} + 8 x^{\frac{4}{3}} \cdot 5^{7} + 5^{8}) d x$

Step 2.2.4.2.3

Rewrite the expression.

$\int x^{\frac{1}{3}} (x^{\frac{32}{3}} + 40 x^{\frac{28}{3}} + 28 x^{4 \cdot 2} \cdot 5^{2} + 56 {(x^{\frac{4}{3}})}^{5} \cdot 5^{3} + 70 {(x^{\frac{4}{3}})}^{4} \cdot 5^{4} + 56 {(x^{\frac{4}{3}})}^{3} \cdot 5^{5} + 28 {(x^{\frac{4}{3}})}^{2} \cdot 5^{6} + 8 x^{\frac{4}{3}} \cdot 5^{7} + 5^{8}) d x$

Step 2.2.4.3

Multiply $4$ by $2$ .

$\int x^{\frac{1}{3}} (x^{\frac{32}{3}} + 40 x^{\frac{28}{3}} + 28 x^{8} \cdot 5^{2} + 56 {(x^{\frac{4}{3}})}^{5} \cdot 5^{3} + 70 {(x^{\frac{4}{3}})}^{4} \cdot 5^{4} + 56 {(x^{\frac{4}{3}})}^{3} \cdot 5^{5} + 28 {(x^{\frac{4}{3}})}^{2} \cdot 5^{6} + 8 x^{\frac{4}{3}} \cdot 5^{7} + 5^{8}) d x$

Step 2.2.5

Raise $5$ to the power of $2$ .

$\int x^{\frac{1}{3}} (x^{\frac{32}{3}} + 40 x^{\frac{28}{3}} + 28 x^{8} \cdot 25 + 56 {(x^{\frac{4}{3}})}^{5} \cdot 5^{3} + 70 {(x^{\frac{4}{3}})}^{4} \cdot 5^{4} + 56 {(x^{\frac{4}{3}})}^{3} \cdot 5^{5} + 28 {(x^{\frac{4}{3}})}^{2} \cdot 5^{6} + 8 x^{\frac{4}{3}} \cdot 5^{7} + 5^{8}) d x$

Step 2.2.6

Multiply $25$ by $28$ .

$\int x^{\frac{1}{3}} (x^{\frac{32}{3}} + 40 x^{\frac{28}{3}} + 700 x^{8} + 56 {(x^{\frac{4}{3}})}^{5} \cdot 5^{3} + 70 {(x^{\frac{4}{3}})}^{4} \cdot 5^{4} + 56 {(x^{\frac{4}{3}})}^{3} \cdot 5^{5} + 28 {(x^{\frac{4}{3}})}^{2} \cdot 5^{6} + 8 x^{\frac{4}{3}} \cdot 5^{7} + 5^{8}) d x$

Step 2.2.7

Multiply the exponents in ${(x^{\frac{4}{3}})}^{5}$ .

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Step 2.2.7.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\int x^{\frac{1}{3}} (x^{\frac{32}{3}} + 40 x^{\frac{28}{3}} + 700 x^{8} + 56 x^{\frac{4}{3} \cdot 5} \cdot 5^{3} + 70 {(x^{\frac{4}{3}})}^{4} \cdot 5^{4} + 56 {(x^{\frac{4}{3}})}^{3} \cdot 5^{5} + 28 {(x^{\frac{4}{3}})}^{2} \cdot 5^{6} + 8 x^{\frac{4}{3}} \cdot 5^{7} + 5^{8}) d x$

Step 2.2.7.2

Multiply $\frac{4}{3} \cdot 5$ .

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Step 2.2.7.2.1

Combine $\frac{4}{3}$ and $5$ .

$\int x^{\frac{1}{3}} (x^{\frac{32}{3}} + 40 x^{\frac{28}{3}} + 700 x^{8} + 56 x^{\frac{4 \cdot 5}{3}} \cdot 5^{3} + 70 {(x^{\frac{4}{3}})}^{4} \cdot 5^{4} + 56 {(x^{\frac{4}{3}})}^{3} \cdot 5^{5} + 28 {(x^{\frac{4}{3}})}^{2} \cdot 5^{6} + 8 x^{\frac{4}{3}} \cdot 5^{7} + 5^{8}) d x$

Step 2.2.7.2.2

Multiply $4$ by $5$ .

$\int x^{\frac{1}{3}} (x^{\frac{32}{3}} + 40 x^{\frac{28}{3}} + 700 x^{8} + 56 x^{\frac{20}{3}} \cdot 5^{3} + 70 {(x^{\frac{4}{3}})}^{4} \cdot 5^{4} + 56 {(x^{\frac{4}{3}})}^{3} \cdot 5^{5} + 28 {(x^{\frac{4}{3}})}^{2} \cdot 5^{6} + 8 x^{\frac{4}{3}} \cdot 5^{7} + 5^{8}) d x$

Step 2.2.8

Raise $5$ to the power of $3$ .

$\int x^{\frac{1}{3}} (x^{\frac{32}{3}} + 40 x^{\frac{28}{3}} + 700 x^{8} + 56 x^{\frac{20}{3}} \cdot 125 + 70 {(x^{\frac{4}{3}})}^{4} \cdot 5^{4} + 56 {(x^{\frac{4}{3}})}^{3} \cdot 5^{5} + 28 {(x^{\frac{4}{3}})}^{2} \cdot 5^{6} + 8 x^{\frac{4}{3}} \cdot 5^{7} + 5^{8}) d x$

Step 2.2.9

Multiply $125$ by $56$ .

$\int x^{\frac{1}{3}} (x^{\frac{32}{3}} + 40 x^{\frac{28}{3}} + 700 x^{8} + 7000 x^{\frac{20}{3}} + 70 {(x^{\frac{4}{3}})}^{4} \cdot 5^{4} + 56 {(x^{\frac{4}{3}})}^{3} \cdot 5^{5} + 28 {(x^{\frac{4}{3}})}^{2} \cdot 5^{6} + 8 x^{\frac{4}{3}} \cdot 5^{7} + 5^{8}) d x$

Step 2.2.10

Multiply the exponents in ${(x^{\frac{4}{3}})}^{4}$ .

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Step 2.2.10.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\int x^{\frac{1}{3}} (x^{\frac{32}{3}} + 40 x^{\frac{28}{3}} + 700 x^{8} + 7000 x^{\frac{20}{3}} + 70 x^{\frac{4}{3} \cdot 4} \cdot 5^{4} + 56 {(x^{\frac{4}{3}})}^{3} \cdot 5^{5} + 28 {(x^{\frac{4}{3}})}^{2} \cdot 5^{6} + 8 x^{\frac{4}{3}} \cdot 5^{7} + 5^{8}) d x$

Step 2.2.10.2

Multiply $\frac{4}{3} \cdot 4$ .

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Step 2.2.10.2.1

Combine $\frac{4}{3}$ and $4$ .

$\int x^{\frac{1}{3}} (x^{\frac{32}{3}} + 40 x^{\frac{28}{3}} + 700 x^{8} + 7000 x^{\frac{20}{3}} + 70 x^{\frac{4 \cdot 4}{3}} \cdot 5^{4} + 56 {(x^{\frac{4}{3}})}^{3} \cdot 5^{5} + 28 {(x^{\frac{4}{3}})}^{2} \cdot 5^{6} + 8 x^{\frac{4}{3}} \cdot 5^{7} + 5^{8}) d x$

Step 2.2.10.2.2

Multiply $4$ by $4$ .

$\int x^{\frac{1}{3}} (x^{\frac{32}{3}} + 40 x^{\frac{28}{3}} + 700 x^{8} + 7000 x^{\frac{20}{3}} + 70 x^{\frac{16}{3}} \cdot 5^{4} + 56 {(x^{\frac{4}{3}})}^{3} \cdot 5^{5} + 28 {(x^{\frac{4}{3}})}^{2} \cdot 5^{6} + 8 x^{\frac{4}{3}} \cdot 5^{7} + 5^{8}) d x$

Step 2.2.11

Raise $5$ to the power of $4$ .

$\int x^{\frac{1}{3}} (x^{\frac{32}{3}} + 40 x^{\frac{28}{3}} + 700 x^{8} + 7000 x^{\frac{20}{3}} + 70 x^{\frac{16}{3}} \cdot 625 + 56 {(x^{\frac{4}{3}})}^{3} \cdot 5^{5} + 28 {(x^{\frac{4}{3}})}^{2} \cdot 5^{6} + 8 x^{\frac{4}{3}} \cdot 5^{7} + 5^{8}) d x$

Step 2.2.12

Multiply $625$ by $70$ .

$\int x^{\frac{1}{3}} (x^{\frac{32}{3}} + 40 x^{\frac{28}{3}} + 700 x^{8} + 7000 x^{\frac{20}{3}} + 43750 x^{\frac{16}{3}} + 56 {(x^{\frac{4}{3}})}^{3} \cdot 5^{5} + 28 {(x^{\frac{4}{3}})}^{2} \cdot 5^{6} + 8 x^{\frac{4}{3}} \cdot 5^{7} + 5^{8}) d x$

Step 2.2.13

Multiply the exponents in ${(x^{\frac{4}{3}})}^{3}$ .

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Step 2.2.13.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\int x^{\frac{1}{3}} (x^{\frac{32}{3}} + 40 x^{\frac{28}{3}} + 700 x^{8} + 7000 x^{\frac{20}{3}} + 43750 x^{\frac{16}{3}} + 56 x^{\frac{4}{3} \cdot 3} \cdot 5^{5} + 28 {(x^{\frac{4}{3}})}^{2} \cdot 5^{6} + 8 x^{\frac{4}{3}} \cdot 5^{7} + 5^{8}) d x$

Step 2.2.13.2

Cancel the common factor of $3$ .

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Step 2.2.13.2.1

Cancel the common factor.

Step 2.2.13.2.2

Rewrite the expression.

$\int x^{\frac{1}{3}} (x^{\frac{32}{3}} + 40 x^{\frac{28}{3}} + 700 x^{8} + 7000 x^{\frac{20}{3}} + 43750 x^{\frac{16}{3}} + 56 x^{4} \cdot 5^{5} + 28 {(x^{\frac{4}{3}})}^{2} \cdot 5^{6} + 8 x^{\frac{4}{3}} \cdot 5^{7} + 5^{8}) d x$

Step 2.2.14

Raise $5$ to the power of $5$ .

$\int x^{\frac{1}{3}} (x^{\frac{32}{3}} + 40 x^{\frac{28}{3}} + 700 x^{8} + 7000 x^{\frac{20}{3}} + 43750 x^{\frac{16}{3}} + 56 x^{4} \cdot 3125 + 28 {(x^{\frac{4}{3}})}^{2} \cdot 5^{6} + 8 x^{\frac{4}{3}} \cdot 5^{7} + 5^{8}) d x$

Step 2.2.15

Multiply $3125$ by $56$ .

$\int x^{\frac{1}{3}} (x^{\frac{32}{3}} + 40 x^{\frac{28}{3}} + 700 x^{8} + 7000 x^{\frac{20}{3}} + 43750 x^{\frac{16}{3}} + 175000 x^{4} + 28 {(x^{\frac{4}{3}})}^{2} \cdot 5^{6} + 8 x^{\frac{4}{3}} \cdot 5^{7} + 5^{8}) d x$

Step 2.2.16

Multiply the exponents in ${(x^{\frac{4}{3}})}^{2}$ .

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Step 2.2.16.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\int x^{\frac{1}{3}} (x^{\frac{32}{3}} + 40 x^{\frac{28}{3}} + 700 x^{8} + 7000 x^{\frac{20}{3}} + 43750 x^{\frac{16}{3}} + 175000 x^{4} + 28 x^{\frac{4}{3} \cdot 2} \cdot 5^{6} + 8 x^{\frac{4}{3}} \cdot 5^{7} + 5^{8}) d x$

Step 2.2.16.2

Multiply $\frac{4}{3} \cdot 2$ .

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Step 2.2.16.2.1

Combine $\frac{4}{3}$ and $2$ .

$\int x^{\frac{1}{3}} (x^{\frac{32}{3}} + 40 x^{\frac{28}{3}} + 700 x^{8} + 7000 x^{\frac{20}{3}} + 43750 x^{\frac{16}{3}} + 175000 x^{4} + 28 x^{\frac{4 \cdot 2}{3}} \cdot 5^{6} + 8 x^{\frac{4}{3}} \cdot 5^{7} + 5^{8}) d x$

Step 2.2.16.2.2

Multiply $4$ by $2$ .

$\int x^{\frac{1}{3}} (x^{\frac{32}{3}} + 40 x^{\frac{28}{3}} + 700 x^{8} + 7000 x^{\frac{20}{3}} + 43750 x^{\frac{16}{3}} + 175000 x^{4} + 28 x^{\frac{8}{3}} \cdot 5^{6} + 8 x^{\frac{4}{3}} \cdot 5^{7} + 5^{8}) d x$

Step 2.2.17

Raise $5$ to the power of $6$ .

$\int x^{\frac{1}{3}} (x^{\frac{32}{3}} + 40 x^{\frac{28}{3}} + 700 x^{8} + 7000 x^{\frac{20}{3}} + 43750 x^{\frac{16}{3}} + 175000 x^{4} + 28 x^{\frac{8}{3}} \cdot 15625 + 8 x^{\frac{4}{3}} \cdot 5^{7} + 5^{8}) d x$

Step 2.2.18

Multiply $15625$ by $28$ .

$\int x^{\frac{1}{3}} (x^{\frac{32}{3}} + 40 x^{\frac{28}{3}} + 700 x^{8} + 7000 x^{\frac{20}{3}} + 43750 x^{\frac{16}{3}} + 175000 x^{4} + 437500 x^{\frac{8}{3}} + 8 x^{\frac{4}{3}} \cdot 5^{7} + 5^{8}) d x$

Step 2.2.19

Raise $5$ to the power of $7$ .

Step 2.2.20

Multiply $78125$ by $8$ .

$\int x^{\frac{1}{3}} (x^{\frac{32}{3}} + 40 x^{\frac{28}{3}} + 700 x^{8} + 7000 x^{\frac{20}{3}} + 43750 x^{\frac{16}{3}} + 175000 x^{4} + 437500 x^{\frac{8}{3}} + 625000 x^{\frac{4}{3}} + 5^{8}) d x$

Step 2.2.21

Raise $5$ to the power of $8$ .

Step 2.3

Apply the distributive property.

$\int x^{\frac{1}{3}} x^{\frac{32}{3}} + x^{\frac{1}{3}} (40 x^{\frac{28}{3}}) + x^{\frac{1}{3}} (700 x^{8}) + x^{\frac{1}{3}} (7000 x^{\frac{20}{3}}) + x^{\frac{1}{3}} (43750 x^{\frac{16}{3}}) + x^{\frac{1}{3}} (175000 x^{4}) + x^{\frac{1}{3}} (437500 x^{\frac{8}{3}}) + x^{\frac{1}{3}} (625000 x^{\frac{4}{3}}) + x^{\frac{1}{3}} \cdot 390625 d x$

Step 2.4

Simplify.

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Step 2.4.1

Multiply $x^{\frac{1}{3}}$ by $x^{\frac{32}{3}}$ by adding the exponents.

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Step 2.4.1.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int x^{\frac{1}{3} + \frac{32}{3}} + x^{\frac{1}{3}} (40 x^{\frac{28}{3}}) + x^{\frac{1}{3}} (700 x^{8}) + x^{\frac{1}{3}} (7000 x^{\frac{20}{3}}) + x^{\frac{1}{3}} (43750 x^{\frac{16}{3}}) + x^{\frac{1}{3}} (175000 x^{4}) + x^{\frac{1}{3}} (437500 x^{\frac{8}{3}}) + x^{\frac{1}{3}} (625000 x^{\frac{4}{3}}) + x^{\frac{1}{3}} \cdot 390625 d x$

Step 2.4.1.2

Combine the numerators over the common denominator.

$\int x^{\frac{1 + 32}{3}} + x^{\frac{1}{3}} (40 x^{\frac{28}{3}}) + x^{\frac{1}{3}} (700 x^{8}) + x^{\frac{1}{3}} (7000 x^{\frac{20}{3}}) + x^{\frac{1}{3}} (43750 x^{\frac{16}{3}}) + x^{\frac{1}{3}} (175000 x^{4}) + x^{\frac{1}{3}} (437500 x^{\frac{8}{3}}) + x^{\frac{1}{3}} (625000 x^{\frac{4}{3}}) + x^{\frac{1}{3}} \cdot 390625 d x$

Step 2.4.1.3

Add $1$ and $32$ .

$\int x^{\frac{33}{3}} + x^{\frac{1}{3}} (40 x^{\frac{28}{3}}) + x^{\frac{1}{3}} (700 x^{8}) + x^{\frac{1}{3}} (7000 x^{\frac{20}{3}}) + x^{\frac{1}{3}} (43750 x^{\frac{16}{3}}) + x^{\frac{1}{3}} (175000 x^{4}) + x^{\frac{1}{3}} (437500 x^{\frac{8}{3}}) + x^{\frac{1}{3}} (625000 x^{\frac{4}{3}}) + x^{\frac{1}{3}} \cdot 390625 d x$

Step 2.4.1.4

Divide $33$ by $3$ .

$\int x^{11} + x^{\frac{1}{3}} (40 x^{\frac{28}{3}}) + x^{\frac{1}{3}} (700 x^{8}) + x^{\frac{1}{3}} (7000 x^{\frac{20}{3}}) + x^{\frac{1}{3}} (43750 x^{\frac{16}{3}}) + x^{\frac{1}{3}} (175000 x^{4}) + x^{\frac{1}{3}} (437500 x^{\frac{8}{3}}) + x^{\frac{1}{3}} (625000 x^{\frac{4}{3}}) + x^{\frac{1}{3}} \cdot 390625 d x$

Step 2.4.2

Rewrite using the commutative property of multiplication.

$\int x^{11} + 40 x^{\frac{1}{3}} x^{\frac{28}{3}} + x^{\frac{1}{3}} (700 x^{8}) + x^{\frac{1}{3}} (7000 x^{\frac{20}{3}}) + x^{\frac{1}{3}} (43750 x^{\frac{16}{3}}) + x^{\frac{1}{3}} (175000 x^{4}) + x^{\frac{1}{3}} (437500 x^{\frac{8}{3}}) + x^{\frac{1}{3}} (625000 x^{\frac{4}{3}}) + x^{\frac{1}{3}} \cdot 390625 d x$

Step 2.4.3

Rewrite using the commutative property of multiplication.

$\int x^{11} + 40 x^{\frac{1}{3}} x^{\frac{28}{3}} + 700 x^{\frac{1}{3}} x^{8} + x^{\frac{1}{3}} (7000 x^{\frac{20}{3}}) + x^{\frac{1}{3}} (43750 x^{\frac{16}{3}}) + x^{\frac{1}{3}} (175000 x^{4}) + x^{\frac{1}{3}} (437500 x^{\frac{8}{3}}) + x^{\frac{1}{3}} (625000 x^{\frac{4}{3}}) + x^{\frac{1}{3}} \cdot 390625 d x$

Step 2.4.4

Rewrite using the commutative property of multiplication.

$\int x^{11} + 40 x^{\frac{1}{3}} x^{\frac{28}{3}} + 700 x^{\frac{1}{3}} x^{8} + 7000 x^{\frac{1}{3}} x^{\frac{20}{3}} + x^{\frac{1}{3}} (43750 x^{\frac{16}{3}}) + x^{\frac{1}{3}} (175000 x^{4}) + x^{\frac{1}{3}} (437500 x^{\frac{8}{3}}) + x^{\frac{1}{3}} (625000 x^{\frac{4}{3}}) + x^{\frac{1}{3}} \cdot 390625 d x$

Step 2.4.5

Rewrite using the commutative property of multiplication.

$\int x^{11} + 40 x^{\frac{1}{3}} x^{\frac{28}{3}} + 700 x^{\frac{1}{3}} x^{8} + 7000 x^{\frac{1}{3}} x^{\frac{20}{3}} + 43750 x^{\frac{1}{3}} x^{\frac{16}{3}} + x^{\frac{1}{3}} (175000 x^{4}) + x^{\frac{1}{3}} (437500 x^{\frac{8}{3}}) + x^{\frac{1}{3}} (625000 x^{\frac{4}{3}}) + x^{\frac{1}{3}} \cdot 390625 d x$

Step 2.4.6

Rewrite using the commutative property of multiplication.

$\int x^{11} + 40 x^{\frac{1}{3}} x^{\frac{28}{3}} + 700 x^{\frac{1}{3}} x^{8} + 7000 x^{\frac{1}{3}} x^{\frac{20}{3}} + 43750 x^{\frac{1}{3}} x^{\frac{16}{3}} + 175000 x^{\frac{1}{3}} x^{4} + x^{\frac{1}{3}} (437500 x^{\frac{8}{3}}) + x^{\frac{1}{3}} (625000 x^{\frac{4}{3}}) + x^{\frac{1}{3}} \cdot 390625 d x$

Step 2.4.7

Rewrite using the commutative property of multiplication.

$\int x^{11} + 40 x^{\frac{1}{3}} x^{\frac{28}{3}} + 700 x^{\frac{1}{3}} x^{8} + 7000 x^{\frac{1}{3}} x^{\frac{20}{3}} + 43750 x^{\frac{1}{3}} x^{\frac{16}{3}} + 175000 x^{\frac{1}{3}} x^{4} + 437500 x^{\frac{1}{3}} x^{\frac{8}{3}} + x^{\frac{1}{3}} (625000 x^{\frac{4}{3}}) + x^{\frac{1}{3}} \cdot 390625 d x$

Step 2.4.8

Rewrite using the commutative property of multiplication.

$\int x^{11} + 40 x^{\frac{1}{3}} x^{\frac{28}{3}} + 700 x^{\frac{1}{3}} x^{8} + 7000 x^{\frac{1}{3}} x^{\frac{20}{3}} + 43750 x^{\frac{1}{3}} x^{\frac{16}{3}} + 175000 x^{\frac{1}{3}} x^{4} + 437500 x^{\frac{1}{3}} x^{\frac{8}{3}} + 625000 x^{\frac{1}{3}} x^{\frac{4}{3}} + x^{\frac{1}{3}} \cdot 390625 d x$

Step 2.4.9

Move $390625$ to the left of $x^{\frac{1}{3}}$ .

$\int x^{11} + 40 x^{\frac{1}{3}} x^{\frac{28}{3}} + 700 x^{\frac{1}{3}} x^{8} + 7000 x^{\frac{1}{3}} x^{\frac{20}{3}} + 43750 x^{\frac{1}{3}} x^{\frac{16}{3}} + 175000 x^{\frac{1}{3}} x^{4} + 437500 x^{\frac{1}{3}} x^{\frac{8}{3}} + 625000 x^{\frac{1}{3}} x^{\frac{4}{3}} + 390625 \cdot x^{\frac{1}{3}} d x$

Step 2.5

Simplify each term.

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Step 2.5.1

Multiply $x^{\frac{1}{3}}$ by $x^{\frac{28}{3}}$ by adding the exponents.

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Step 2.5.1.1

Move $x^{\frac{28}{3}}$ .

$\int x^{11} + 40 (x^{\frac{28}{3}} x^{\frac{1}{3}}) + 700 x^{\frac{1}{3}} x^{8} + 7000 x^{\frac{1}{3}} x^{\frac{20}{3}} + 43750 x^{\frac{1}{3}} x^{\frac{16}{3}} + 175000 x^{\frac{1}{3}} x^{4} + 437500 x^{\frac{1}{3}} x^{\frac{8}{3}} + 625000 x^{\frac{1}{3}} x^{\frac{4}{3}} + 390625 \cdot x^{\frac{1}{3}} d x$

Step 2.5.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int x^{11} + 40 x^{\frac{28}{3} + \frac{1}{3}} + 700 x^{\frac{1}{3}} x^{8} + 7000 x^{\frac{1}{3}} x^{\frac{20}{3}} + 43750 x^{\frac{1}{3}} x^{\frac{16}{3}} + 175000 x^{\frac{1}{3}} x^{4} + 437500 x^{\frac{1}{3}} x^{\frac{8}{3}} + 625000 x^{\frac{1}{3}} x^{\frac{4}{3}} + 390625 \cdot x^{\frac{1}{3}} d x$

Step 2.5.1.3

Combine the numerators over the common denominator.

$\int x^{11} + 40 x^{\frac{28 + 1}{3}} + 700 x^{\frac{1}{3}} x^{8} + 7000 x^{\frac{1}{3}} x^{\frac{20}{3}} + 43750 x^{\frac{1}{3}} x^{\frac{16}{3}} + 175000 x^{\frac{1}{3}} x^{4} + 437500 x^{\frac{1}{3}} x^{\frac{8}{3}} + 625000 x^{\frac{1}{3}} x^{\frac{4}{3}} + 390625 \cdot x^{\frac{1}{3}} d x$

Step 2.5.1.4

Add $28$ and $1$ .

$\int x^{11} + 40 x^{\frac{29}{3}} + 700 x^{\frac{1}{3}} x^{8} + 7000 x^{\frac{1}{3}} x^{\frac{20}{3}} + 43750 x^{\frac{1}{3}} x^{\frac{16}{3}} + 175000 x^{\frac{1}{3}} x^{4} + 437500 x^{\frac{1}{3}} x^{\frac{8}{3}} + 625000 x^{\frac{1}{3}} x^{\frac{4}{3}} + 390625 \cdot x^{\frac{1}{3}} d x$

Step 2.5.2

Multiply $x^{\frac{1}{3}}$ by $x^{8}$ by adding the exponents.

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Step 2.5.2.1

Move $x^{8}$ .

$\int x^{11} + 40 x^{\frac{29}{3}} + 700 (x^{8} x^{\frac{1}{3}}) + 7000 x^{\frac{1}{3}} x^{\frac{20}{3}} + 43750 x^{\frac{1}{3}} x^{\frac{16}{3}} + 175000 x^{\frac{1}{3}} x^{4} + 437500 x^{\frac{1}{3}} x^{\frac{8}{3}} + 625000 x^{\frac{1}{3}} x^{\frac{4}{3}} + 390625 \cdot x^{\frac{1}{3}} d x$

Step 2.5.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int x^{11} + 40 x^{\frac{29}{3}} + 700 x^{8 + \frac{1}{3}} + 7000 x^{\frac{1}{3}} x^{\frac{20}{3}} + 43750 x^{\frac{1}{3}} x^{\frac{16}{3}} + 175000 x^{\frac{1}{3}} x^{4} + 437500 x^{\frac{1}{3}} x^{\frac{8}{3}} + 625000 x^{\frac{1}{3}} x^{\frac{4}{3}} + 390625 \cdot x^{\frac{1}{3}} d x$

Step 2.5.2.3

To write $8$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\int x^{11} + 40 x^{\frac{29}{3}} + 700 x^{8 \cdot \frac{3}{3} + \frac{1}{3}} + 7000 x^{\frac{1}{3}} x^{\frac{20}{3}} + 43750 x^{\frac{1}{3}} x^{\frac{16}{3}} + 175000 x^{\frac{1}{3}} x^{4} + 437500 x^{\frac{1}{3}} x^{\frac{8}{3}} + 625000 x^{\frac{1}{3}} x^{\frac{4}{3}} + 390625 \cdot x^{\frac{1}{3}} d x$

Step 2.5.2.4

Combine $8$ and $\frac{3}{3}$ .

$\int x^{11} + 40 x^{\frac{29}{3}} + 700 x^{\frac{8 \cdot 3}{3} + \frac{1}{3}} + 7000 x^{\frac{1}{3}} x^{\frac{20}{3}} + 43750 x^{\frac{1}{3}} x^{\frac{16}{3}} + 175000 x^{\frac{1}{3}} x^{4} + 437500 x^{\frac{1}{3}} x^{\frac{8}{3}} + 625000 x^{\frac{1}{3}} x^{\frac{4}{3}} + 390625 \cdot x^{\frac{1}{3}} d x$

Step 2.5.2.5

Combine the numerators over the common denominator.

$\int x^{11} + 40 x^{\frac{29}{3}} + 700 x^{\frac{8 \cdot 3 + 1}{3}} + 7000 x^{\frac{1}{3}} x^{\frac{20}{3}} + 43750 x^{\frac{1}{3}} x^{\frac{16}{3}} + 175000 x^{\frac{1}{3}} x^{4} + 437500 x^{\frac{1}{3}} x^{\frac{8}{3}} + 625000 x^{\frac{1}{3}} x^{\frac{4}{3}} + 390625 \cdot x^{\frac{1}{3}} d x$

Step 2.5.2.6

Simplify the numerator.

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Step 2.5.2.6.1

Multiply $8$ by $3$ .

$\int x^{11} + 40 x^{\frac{29}{3}} + 700 x^{\frac{24 + 1}{3}} + 7000 x^{\frac{1}{3}} x^{\frac{20}{3}} + 43750 x^{\frac{1}{3}} x^{\frac{16}{3}} + 175000 x^{\frac{1}{3}} x^{4} + 437500 x^{\frac{1}{3}} x^{\frac{8}{3}} + 625000 x^{\frac{1}{3}} x^{\frac{4}{3}} + 390625 \cdot x^{\frac{1}{3}} d x$

Step 2.5.2.6.2

Add $24$ and $1$ .

$\int x^{11} + 40 x^{\frac{29}{3}} + 700 x^{\frac{25}{3}} + 7000 x^{\frac{1}{3}} x^{\frac{20}{3}} + 43750 x^{\frac{1}{3}} x^{\frac{16}{3}} + 175000 x^{\frac{1}{3}} x^{4} + 437500 x^{\frac{1}{3}} x^{\frac{8}{3}} + 625000 x^{\frac{1}{3}} x^{\frac{4}{3}} + 390625 \cdot x^{\frac{1}{3}} d x$

Step 2.5.3

Multiply $x^{\frac{1}{3}}$ by $x^{\frac{20}{3}}$ by adding the exponents.

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Step 2.5.3.1

Move $x^{\frac{20}{3}}$ .

$\int x^{11} + 40 x^{\frac{29}{3}} + 700 x^{\frac{25}{3}} + 7000 (x^{\frac{20}{3}} x^{\frac{1}{3}}) + 43750 x^{\frac{1}{3}} x^{\frac{16}{3}} + 175000 x^{\frac{1}{3}} x^{4} + 437500 x^{\frac{1}{3}} x^{\frac{8}{3}} + 625000 x^{\frac{1}{3}} x^{\frac{4}{3}} + 390625 \cdot x^{\frac{1}{3}} d x$

Step 2.5.3.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int x^{11} + 40 x^{\frac{29}{3}} + 700 x^{\frac{25}{3}} + 7000 x^{\frac{20}{3} + \frac{1}{3}} + 43750 x^{\frac{1}{3}} x^{\frac{16}{3}} + 175000 x^{\frac{1}{3}} x^{4} + 437500 x^{\frac{1}{3}} x^{\frac{8}{3}} + 625000 x^{\frac{1}{3}} x^{\frac{4}{3}} + 390625 \cdot x^{\frac{1}{3}} d x$

Step 2.5.3.3

Combine the numerators over the common denominator.

$\int x^{11} + 40 x^{\frac{29}{3}} + 700 x^{\frac{25}{3}} + 7000 x^{\frac{20 + 1}{3}} + 43750 x^{\frac{1}{3}} x^{\frac{16}{3}} + 175000 x^{\frac{1}{3}} x^{4} + 437500 x^{\frac{1}{3}} x^{\frac{8}{3}} + 625000 x^{\frac{1}{3}} x^{\frac{4}{3}} + 390625 \cdot x^{\frac{1}{3}} d x$

Step 2.5.3.4

Add $20$ and $1$ .

$\int x^{11} + 40 x^{\frac{29}{3}} + 700 x^{\frac{25}{3}} + 7000 x^{\frac{21}{3}} + 43750 x^{\frac{1}{3}} x^{\frac{16}{3}} + 175000 x^{\frac{1}{3}} x^{4} + 437500 x^{\frac{1}{3}} x^{\frac{8}{3}} + 625000 x^{\frac{1}{3}} x^{\frac{4}{3}} + 390625 \cdot x^{\frac{1}{3}} d x$

Step 2.5.3.5

Divide $21$ by $3$ .

$\int x^{11} + 40 x^{\frac{29}{3}} + 700 x^{\frac{25}{3}} + 7000 x^{7} + 43750 x^{\frac{1}{3}} x^{\frac{16}{3}} + 175000 x^{\frac{1}{3}} x^{4} + 437500 x^{\frac{1}{3}} x^{\frac{8}{3}} + 625000 x^{\frac{1}{3}} x^{\frac{4}{3}} + 390625 \cdot x^{\frac{1}{3}} d x$

Step 2.5.4

Multiply $x^{\frac{1}{3}}$ by $x^{\frac{16}{3}}$ by adding the exponents.

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Step 2.5.4.1

Move $x^{\frac{16}{3}}$ .

$\int x^{11} + 40 x^{\frac{29}{3}} + 700 x^{\frac{25}{3}} + 7000 x^{7} + 43750 (x^{\frac{16}{3}} x^{\frac{1}{3}}) + 175000 x^{\frac{1}{3}} x^{4} + 437500 x^{\frac{1}{3}} x^{\frac{8}{3}} + 625000 x^{\frac{1}{3}} x^{\frac{4}{3}} + 390625 \cdot x^{\frac{1}{3}} d x$

Step 2.5.4.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int x^{11} + 40 x^{\frac{29}{3}} + 700 x^{\frac{25}{3}} + 7000 x^{7} + 43750 x^{\frac{16}{3} + \frac{1}{3}} + 175000 x^{\frac{1}{3}} x^{4} + 437500 x^{\frac{1}{3}} x^{\frac{8}{3}} + 625000 x^{\frac{1}{3}} x^{\frac{4}{3}} + 390625 \cdot x^{\frac{1}{3}} d x$

Step 2.5.4.3

Combine the numerators over the common denominator.

$\int x^{11} + 40 x^{\frac{29}{3}} + 700 x^{\frac{25}{3}} + 7000 x^{7} + 43750 x^{\frac{16 + 1}{3}} + 175000 x^{\frac{1}{3}} x^{4} + 437500 x^{\frac{1}{3}} x^{\frac{8}{3}} + 625000 x^{\frac{1}{3}} x^{\frac{4}{3}} + 390625 \cdot x^{\frac{1}{3}} d x$

Step 2.5.4.4

Add $16$ and $1$ .

$\int x^{11} + 40 x^{\frac{29}{3}} + 700 x^{\frac{25}{3}} + 7000 x^{7} + 43750 x^{\frac{17}{3}} + 175000 x^{\frac{1}{3}} x^{4} + 437500 x^{\frac{1}{3}} x^{\frac{8}{3}} + 625000 x^{\frac{1}{3}} x^{\frac{4}{3}} + 390625 \cdot x^{\frac{1}{3}} d x$

Step 2.5.5

Multiply $x^{\frac{1}{3}}$ by $x^{4}$ by adding the exponents.

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Step 2.5.5.1

Move $x^{4}$ .

$\int x^{11} + 40 x^{\frac{29}{3}} + 700 x^{\frac{25}{3}} + 7000 x^{7} + 43750 x^{\frac{17}{3}} + 175000 (x^{4} x^{\frac{1}{3}}) + 437500 x^{\frac{1}{3}} x^{\frac{8}{3}} + 625000 x^{\frac{1}{3}} x^{\frac{4}{3}} + 390625 \cdot x^{\frac{1}{3}} d x$

Step 2.5.5.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int x^{11} + 40 x^{\frac{29}{3}} + 700 x^{\frac{25}{3}} + 7000 x^{7} + 43750 x^{\frac{17}{3}} + 175000 x^{4 + \frac{1}{3}} + 437500 x^{\frac{1}{3}} x^{\frac{8}{3}} + 625000 x^{\frac{1}{3}} x^{\frac{4}{3}} + 390625 \cdot x^{\frac{1}{3}} d x$

Step 2.5.5.3

To write $4$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\int x^{11} + 40 x^{\frac{29}{3}} + 700 x^{\frac{25}{3}} + 7000 x^{7} + 43750 x^{\frac{17}{3}} + 175000 x^{4 \cdot \frac{3}{3} + \frac{1}{3}} + 437500 x^{\frac{1}{3}} x^{\frac{8}{3}} + 625000 x^{\frac{1}{3}} x^{\frac{4}{3}} + 390625 \cdot x^{\frac{1}{3}} d x$

Step 2.5.5.4

Combine $4$ and $\frac{3}{3}$ .

$\int x^{11} + 40 x^{\frac{29}{3}} + 700 x^{\frac{25}{3}} + 7000 x^{7} + 43750 x^{\frac{17}{3}} + 175000 x^{\frac{4 \cdot 3}{3} + \frac{1}{3}} + 437500 x^{\frac{1}{3}} x^{\frac{8}{3}} + 625000 x^{\frac{1}{3}} x^{\frac{4}{3}} + 390625 \cdot x^{\frac{1}{3}} d x$

Step 2.5.5.5

Combine the numerators over the common denominator.

$\int x^{11} + 40 x^{\frac{29}{3}} + 700 x^{\frac{25}{3}} + 7000 x^{7} + 43750 x^{\frac{17}{3}} + 175000 x^{\frac{4 \cdot 3 + 1}{3}} + 437500 x^{\frac{1}{3}} x^{\frac{8}{3}} + 625000 x^{\frac{1}{3}} x^{\frac{4}{3}} + 390625 \cdot x^{\frac{1}{3}} d x$

Step 2.5.5.6

Simplify the numerator.

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Step 2.5.5.6.1

Multiply $4$ by $3$ .

$\int x^{11} + 40 x^{\frac{29}{3}} + 700 x^{\frac{25}{3}} + 7000 x^{7} + 43750 x^{\frac{17}{3}} + 175000 x^{\frac{12 + 1}{3}} + 437500 x^{\frac{1}{3}} x^{\frac{8}{3}} + 625000 x^{\frac{1}{3}} x^{\frac{4}{3}} + 390625 \cdot x^{\frac{1}{3}} d x$

Step 2.5.5.6.2

Add $12$ and $1$ .

$\int x^{11} + 40 x^{\frac{29}{3}} + 700 x^{\frac{25}{3}} + 7000 x^{7} + 43750 x^{\frac{17}{3}} + 175000 x^{\frac{13}{3}} + 437500 x^{\frac{1}{3}} x^{\frac{8}{3}} + 625000 x^{\frac{1}{3}} x^{\frac{4}{3}} + 390625 \cdot x^{\frac{1}{3}} d x$

Step 2.5.6

Multiply $x^{\frac{1}{3}}$ by $x^{\frac{8}{3}}$ by adding the exponents.

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Step 2.5.6.1

Move $x^{\frac{8}{3}}$ .

$\int x^{11} + 40 x^{\frac{29}{3}} + 700 x^{\frac{25}{3}} + 7000 x^{7} + 43750 x^{\frac{17}{3}} + 175000 x^{\frac{13}{3}} + 437500 (x^{\frac{8}{3}} x^{\frac{1}{3}}) + 625000 x^{\frac{1}{3}} x^{\frac{4}{3}} + 390625 \cdot x^{\frac{1}{3}} d x$

Step 2.5.6.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int x^{11} + 40 x^{\frac{29}{3}} + 700 x^{\frac{25}{3}} + 7000 x^{7} + 43750 x^{\frac{17}{3}} + 175000 x^{\frac{13}{3}} + 437500 x^{\frac{8}{3} + \frac{1}{3}} + 625000 x^{\frac{1}{3}} x^{\frac{4}{3}} + 390625 \cdot x^{\frac{1}{3}} d x$

Step 2.5.6.3

Combine the numerators over the common denominator.

$\int x^{11} + 40 x^{\frac{29}{3}} + 700 x^{\frac{25}{3}} + 7000 x^{7} + 43750 x^{\frac{17}{3}} + 175000 x^{\frac{13}{3}} + 437500 x^{\frac{8 + 1}{3}} + 625000 x^{\frac{1}{3}} x^{\frac{4}{3}} + 390625 \cdot x^{\frac{1}{3}} d x$

Step 2.5.6.4

Add $8$ and $1$ .

$\int x^{11} + 40 x^{\frac{29}{3}} + 700 x^{\frac{25}{3}} + 7000 x^{7} + 43750 x^{\frac{17}{3}} + 175000 x^{\frac{13}{3}} + 437500 x^{\frac{9}{3}} + 625000 x^{\frac{1}{3}} x^{\frac{4}{3}} + 390625 \cdot x^{\frac{1}{3}} d x$

Step 2.5.6.5

Divide $9$ by $3$ .

$\int x^{11} + 40 x^{\frac{29}{3}} + 700 x^{\frac{25}{3}} + 7000 x^{7} + 43750 x^{\frac{17}{3}} + 175000 x^{\frac{13}{3}} + 437500 x^{3} + 625000 x^{\frac{1}{3}} x^{\frac{4}{3}} + 390625 \cdot x^{\frac{1}{3}} d x$

Step 2.5.7

Multiply $x^{\frac{1}{3}}$ by $x^{\frac{4}{3}}$ by adding the exponents.

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Step 2.5.7.1

Move $x^{\frac{4}{3}}$ .

$\int x^{11} + 40 x^{\frac{29}{3}} + 700 x^{\frac{25}{3}} + 7000 x^{7} + 43750 x^{\frac{17}{3}} + 175000 x^{\frac{13}{3}} + 437500 x^{3} + 625000 (x^{\frac{4}{3}} x^{\frac{1}{3}}) + 390625 \cdot x^{\frac{1}{3}} d x$

Step 2.5.7.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int x^{11} + 40 x^{\frac{29}{3}} + 700 x^{\frac{25}{3}} + 7000 x^{7} + 43750 x^{\frac{17}{3}} + 175000 x^{\frac{13}{3}} + 437500 x^{3} + 625000 x^{\frac{4}{3} + \frac{1}{3}} + 390625 \cdot x^{\frac{1}{3}} d x$

Step 2.5.7.3

Combine the numerators over the common denominator.

$\int x^{11} + 40 x^{\frac{29}{3}} + 700 x^{\frac{25}{3}} + 7000 x^{7} + 43750 x^{\frac{17}{3}} + 175000 x^{\frac{13}{3}} + 437500 x^{3} + 625000 x^{\frac{4 + 1}{3}} + 390625 \cdot x^{\frac{1}{3}} d x$

Step 2.5.7.4

Add $4$ and $1$ .

$\int x^{11} + 40 x^{\frac{29}{3}} + 700 x^{\frac{25}{3}} + 7000 x^{7} + 43750 x^{\frac{17}{3}} + 175000 x^{\frac{13}{3}} + 437500 x^{3} + 625000 x^{\frac{5}{3}} + 390625 \cdot x^{\frac{1}{3}} d x$

$\int x^{11} + 40 x^{\frac{29}{3}} + 700 x^{\frac{25}{3}} + 7000 x^{7} + 43750 x^{\frac{17}{3}} + 175000 x^{\frac{13}{3}} + 437500 x^{3} + 625000 x^{\frac{5}{3}} + 390625 x^{\frac{1}{3}} d x$

Step 3

Split the single integral into multiple integrals.

$\int x^{11} d x + \int 40 x^{\frac{29}{3}} d x + \int 700 x^{\frac{25}{3}} d x + \int 7000 x^{7} d x + \int 43750 x^{\frac{17}{3}} d x + \int 175000 x^{\frac{13}{3}} d x + \int 437500 x^{3} d x + \int 625000 x^{\frac{5}{3}} d x + \int 390625 x^{\frac{1}{3}} d x$

Step 4

By the Power Rule, the integral of $x^{11}$ with respect to $x$ is $\frac{1}{12} x^{12}$ .

$\frac{1}{12} x^{12} + C + \int 40 x^{\frac{29}{3}} d x + \int 700 x^{\frac{25}{3}} d x + \int 7000 x^{7} d x + \int 43750 x^{\frac{17}{3}} d x + \int 175000 x^{\frac{13}{3}} d x + \int 437500 x^{3} d x + \int 625000 x^{\frac{5}{3}} d x + \int 390625 x^{\frac{1}{3}} d x$

Step 5

Since $40$ is constant with respect to $x$ , move $40$ out of the integral.

$\frac{1}{12} x^{12} + C + 40 \int x^{\frac{29}{3}} d x + \int 700 x^{\frac{25}{3}} d x + \int 7000 x^{7} d x + \int 43750 x^{\frac{17}{3}} d x + \int 175000 x^{\frac{13}{3}} d x + \int 437500 x^{3} d x + \int 625000 x^{\frac{5}{3}} d x + \int 390625 x^{\frac{1}{3}} d x$

Step 6

By the Power Rule, the integral of $x^{\frac{29}{3}}$ with respect to $x$ is $\frac{3}{32} x^{\frac{32}{3}}$ .

$\frac{1}{12} x^{12} + C + 40 (\frac{3}{32} x^{\frac{32}{3}} + C) + \int 700 x^{\frac{25}{3}} d x + \int 7000 x^{7} d x + \int 43750 x^{\frac{17}{3}} d x + \int 175000 x^{\frac{13}{3}} d x + \int 437500 x^{3} d x + \int 625000 x^{\frac{5}{3}} d x + \int 390625 x^{\frac{1}{3}} d x$

Step 7

Since $700$ is constant with respect to $x$ , move $700$ out of the integral.

$\frac{1}{12} x^{12} + C + 40 (\frac{3}{32} x^{\frac{32}{3}} + C) + 700 \int x^{\frac{25}{3}} d x + \int 7000 x^{7} d x + \int 43750 x^{\frac{17}{3}} d x + \int 175000 x^{\frac{13}{3}} d x + \int 437500 x^{3} d x + \int 625000 x^{\frac{5}{3}} d x + \int 390625 x^{\frac{1}{3}} d x$

Step 8

By the Power Rule, the integral of $x^{\frac{25}{3}}$ with respect to $x$ is $\frac{3}{28} x^{\frac{28}{3}}$ .

$\frac{1}{12} x^{12} + C + 40 (\frac{3}{32} x^{\frac{32}{3}} + C) + 700 (\frac{3}{28} x^{\frac{28}{3}} + C) + \int 7000 x^{7} d x + \int 43750 x^{\frac{17}{3}} d x + \int 175000 x^{\frac{13}{3}} d x + \int 437500 x^{3} d x + \int 625000 x^{\frac{5}{3}} d x + \int 390625 x^{\frac{1}{3}} d x$

Step 9

Since $7000$ is constant with respect to $x$ , move $7000$ out of the integral.

$\frac{1}{12} x^{12} + C + 40 (\frac{3}{32} x^{\frac{32}{3}} + C) + 700 (\frac{3}{28} x^{\frac{28}{3}} + C) + 7000 \int x^{7} d x + \int 43750 x^{\frac{17}{3}} d x + \int 175000 x^{\frac{13}{3}} d x + \int 437500 x^{3} d x + \int 625000 x^{\frac{5}{3}} d x + \int 390625 x^{\frac{1}{3}} d x$

Step 10

By the Power Rule, the integral of $x^{7}$ with respect to $x$ is $\frac{1}{8} x^{8}$ .

$\frac{1}{12} x^{12} + C + 40 (\frac{3}{32} x^{\frac{32}{3}} + C) + 700 (\frac{3}{28} x^{\frac{28}{3}} + C) + 7000 (\frac{1}{8} x^{8} + C) + \int 43750 x^{\frac{17}{3}} d x + \int 175000 x^{\frac{13}{3}} d x + \int 437500 x^{3} d x + \int 625000 x^{\frac{5}{3}} d x + \int 390625 x^{\frac{1}{3}} d x$

Step 11

Since $43750$ is constant with respect to $x$ , move $43750$ out of the integral.

$\frac{1}{12} x^{12} + C + 40 (\frac{3}{32} x^{\frac{32}{3}} + C) + 700 (\frac{3}{28} x^{\frac{28}{3}} + C) + 7000 (\frac{1}{8} x^{8} + C) + 43750 \int x^{\frac{17}{3}} d x + \int 175000 x^{\frac{13}{3}} d x + \int 437500 x^{3} d x + \int 625000 x^{\frac{5}{3}} d x + \int 390625 x^{\frac{1}{3}} d x$

Step 12

By the Power Rule, the integral of $x^{\frac{17}{3}}$ with respect to $x$ is $\frac{3}{20} x^{\frac{20}{3}}$ .

$\frac{1}{12} x^{12} + C + 40 (\frac{3}{32} x^{\frac{32}{3}} + C) + 700 (\frac{3}{28} x^{\frac{28}{3}} + C) + 7000 (\frac{1}{8} x^{8} + C) + 43750 (\frac{3}{20} x^{\frac{20}{3}} + C) + \int 175000 x^{\frac{13}{3}} d x + \int 437500 x^{3} d x + \int 625000 x^{\frac{5}{3}} d x + \int 390625 x^{\frac{1}{3}} d x$

Step 13

Since $175000$ is constant with respect to $x$ , move $175000$ out of the integral.

$\frac{1}{12} x^{12} + C + 40 (\frac{3}{32} x^{\frac{32}{3}} + C) + 700 (\frac{3}{28} x^{\frac{28}{3}} + C) + 7000 (\frac{1}{8} x^{8} + C) + 43750 (\frac{3}{20} x^{\frac{20}{3}} + C) + 175000 \int x^{\frac{13}{3}} d x + \int 437500 x^{3} d x + \int 625000 x^{\frac{5}{3}} d x + \int 390625 x^{\frac{1}{3}} d x$

Step 14

By the Power Rule, the integral of $x^{\frac{13}{3}}$ with respect to $x$ is $\frac{3}{16} x^{\frac{16}{3}}$ .

$\frac{1}{12} x^{12} + C + 40 (\frac{3}{32} x^{\frac{32}{3}} + C) + 700 (\frac{3}{28} x^{\frac{28}{3}} + C) + 7000 (\frac{1}{8} x^{8} + C) + 43750 (\frac{3}{20} x^{\frac{20}{3}} + C) + 175000 (\frac{3}{16} x^{\frac{16}{3}} + C) + \int 437500 x^{3} d x + \int 625000 x^{\frac{5}{3}} d x + \int 390625 x^{\frac{1}{3}} d x$

Step 15

Since $437500$ is constant with respect to $x$ , move $437500$ out of the integral.

$\frac{1}{12} x^{12} + C + 40 (\frac{3}{32} x^{\frac{32}{3}} + C) + 700 (\frac{3}{28} x^{\frac{28}{3}} + C) + 7000 (\frac{1}{8} x^{8} + C) + 43750 (\frac{3}{20} x^{\frac{20}{3}} + C) + 175000 (\frac{3}{16} x^{\frac{16}{3}} + C) + 437500 \int x^{3} d x + \int 625000 x^{\frac{5}{3}} d x + \int 390625 x^{\frac{1}{3}} d x$

Step 16

By the Power Rule, the integral of $x^{3}$ with respect to $x$ is $\frac{1}{4} x^{4}$ .

$\frac{1}{12} x^{12} + C + 40 (\frac{3}{32} x^{\frac{32}{3}} + C) + 700 (\frac{3}{28} x^{\frac{28}{3}} + C) + 7000 (\frac{1}{8} x^{8} + C) + 43750 (\frac{3}{20} x^{\frac{20}{3}} + C) + 175000 (\frac{3}{16} x^{\frac{16}{3}} + C) + 437500 (\frac{1}{4} x^{4} + C) + \int 625000 x^{\frac{5}{3}} d x + \int 390625 x^{\frac{1}{3}} d x$

Step 17

Since $625000$ is constant with respect to $x$ , move $625000$ out of the integral.

$\frac{1}{12} x^{12} + C + 40 (\frac{3}{32} x^{\frac{32}{3}} + C) + 700 (\frac{3}{28} x^{\frac{28}{3}} + C) + 7000 (\frac{1}{8} x^{8} + C) + 43750 (\frac{3}{20} x^{\frac{20}{3}} + C) + 175000 (\frac{3}{16} x^{\frac{16}{3}} + C) + 437500 (\frac{1}{4} x^{4} + C) + 625000 \int x^{\frac{5}{3}} d x + \int 390625 x^{\frac{1}{3}} d x$

Step 18

By the Power Rule, the integral of $x^{\frac{5}{3}}$ with respect to $x$ is $\frac{3}{8} x^{\frac{8}{3}}$ .

$\frac{1}{12} x^{12} + C + 40 (\frac{3}{32} x^{\frac{32}{3}} + C) + 700 (\frac{3}{28} x^{\frac{28}{3}} + C) + 7000 (\frac{1}{8} x^{8} + C) + 43750 (\frac{3}{20} x^{\frac{20}{3}} + C) + 175000 (\frac{3}{16} x^{\frac{16}{3}} + C) + 437500 (\frac{1}{4} x^{4} + C) + 625000 (\frac{3}{8} x^{\frac{8}{3}} + C) + \int 390625 x^{\frac{1}{3}} d x$

Step 19

Since $390625$ is constant with respect to $x$ , move $390625$ out of the integral.

$\frac{1}{12} x^{12} + C + 40 (\frac{3}{32} x^{\frac{32}{3}} + C) + 700 (\frac{3}{28} x^{\frac{28}{3}} + C) + 7000 (\frac{1}{8} x^{8} + C) + 43750 (\frac{3}{20} x^{\frac{20}{3}} + C) + 175000 (\frac{3}{16} x^{\frac{16}{3}} + C) + 437500 (\frac{1}{4} x^{4} + C) + 625000 (\frac{3}{8} x^{\frac{8}{3}} + C) + 390625 \int x^{\frac{1}{3}} d x$

Step 20

By the Power Rule, the integral of $x^{\frac{1}{3}}$ with respect to $x$ is $\frac{3}{4} x^{\frac{4}{3}}$ .

$\frac{1}{12} x^{12} + C + 40 (\frac{3}{32} x^{\frac{32}{3}} + C) + 700 (\frac{3}{28} x^{\frac{28}{3}} + C) + 7000 (\frac{1}{8} x^{8} + C) + 43750 (\frac{3}{20} x^{\frac{20}{3}} + C) + 175000 (\frac{3}{16} x^{\frac{16}{3}} + C) + 437500 (\frac{1}{4} x^{4} + C) + 625000 (\frac{3}{8} x^{\frac{8}{3}} + C) + 390625 (\frac{3}{4} x^{\frac{4}{3}} + C)$

Step 21

Simplify.

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Step 21.1

Simplify.

$\frac{x^{12}}{12} + \frac{15 x^{\frac{32}{3}}}{4} + 75 x^{\frac{28}{3}} + 875 x^{8} + \frac{13125 x^{\frac{20}{3}}}{2} + \frac{65625 x^{\frac{16}{3}}}{2} + 109375 x^{4} + 234375 x^{\frac{8}{3}} + 390625 (\frac{3}{4} x^{\frac{4}{3}}) + C$

Step 21.2

Simplify.

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Step 21.2.1

Combine $\frac{3}{4}$ and $x^{\frac{4}{3}}$ .

Step 21.2.2

Combine $390625$ and $\frac{3 x^{\frac{4}{3}}}{4}$ .

Step 21.2.3

Multiply $3$ by $390625$ .

Step 21.3

Reorder terms.

$\frac{1}{12} x^{12} + \frac{15}{4} x^{\frac{32}{3}} + 75 x^{\frac{28}{3}} + 875 x^{8} + \frac{13125}{2} x^{\frac{20}{3}} + \frac{65625}{2} x^{\frac{16}{3}} + 109375 x^{4} + 234375 x^{\frac{8}{3}} + \frac{1171875}{4} x^{\frac{4}{3}} + C$