Calculus Examples

$\int_{0}^{1} x (\sqrt[3]{x} + \sqrt[4]{x}) d x$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x}$ as $x^{\frac{1}{3}}$ .

$\int_{0}^{1} x (x^{\frac{1}{3}} + \sqrt[4]{x}) d x$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{x}$ as $x^{\frac{1}{4}}$ .

$\int_{0}^{1} x (x^{\frac{1}{3}} + x^{\frac{1}{4}}) d x$

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

$\int_{0}^{1} x \cdot x^{\frac{1}{3}} + x \cdot x^{\frac{1}{4}} d x$

Step 3.2

Raise $x$ to the power of $1$ .

$\int_{0}^{1} x^{1} x^{\frac{1}{3}} + x \cdot x^{\frac{1}{4}} d x$

Step 3.3

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

$\int_{0}^{1} x^{1 + \frac{1}{3}} + x \cdot x^{\frac{1}{4}} d x$

Step 3.4

Write $1$ as a fraction with a common denominator.

$\int_{0}^{1} x^{\frac{3}{3} + \frac{1}{3}} + x \cdot x^{\frac{1}{4}} d x$

Step 3.5

Combine the numerators over the common denominator.

$\int_{0}^{1} x^{\frac{3 + 1}{3}} + x \cdot x^{\frac{1}{4}} d x$

Step 3.6

Add $3$ and $1$ .

$\int_{0}^{1} x^{\frac{4}{3}} + x \cdot x^{\frac{1}{4}} d x$

Step 3.7

Raise $x$ to the power of $1$ .

$\int_{0}^{1} x^{\frac{4}{3}} + x^{1} x^{\frac{1}{4}} d x$

Step 3.8

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

$\int_{0}^{1} x^{\frac{4}{3}} + x^{1 + \frac{1}{4}} d x$

Step 3.9

Write $1$ as a fraction with a common denominator.

$\int_{0}^{1} x^{\frac{4}{3}} + x^{\frac{4}{4} + \frac{1}{4}} d x$

Step 3.10

Combine the numerators over the common denominator.

$\int_{0}^{1} x^{\frac{4}{3}} + x^{\frac{4 + 1}{4}} d x$

Step 3.11

Add $4$ and $1$ .

$\int_{0}^{1} x^{\frac{4}{3}} + x^{\frac{5}{4}} d x$

Step 4

Split the single integral into multiple integrals.

$\int_{0}^{1} x^{\frac{4}{3}} d x + \int_{0}^{1} x^{\frac{5}{4}} d x$

Step 5

By the Power Rule, the integral of $x^{\frac{4}{3}}$ with respect to $x$ is $\frac{3}{7} x^{\frac{7}{3}}$ .

$\frac{3}{7} x^{\frac{7}{3}}]_{0}^{1} + \int_{0}^{1} x^{\frac{5}{4}} d x$

Step 6

By the Power Rule, the integral of $x^{\frac{5}{4}}$ with respect to $x$ is $\frac{4}{9} x^{\frac{9}{4}}$ .

$\frac{3}{7} x^{\frac{7}{3}}]_{0}^{1} + \frac{4}{9} x^{\frac{9}{4}}]_{0}^{1}$

Step 7

Simplify the answer.

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Step 7.1

Combine $\frac{3}{7} x^{\frac{7}{3}}]_{0}^{1}$ and $\frac{4}{9} x^{\frac{9}{4}}]_{0}^{1}$ .

$\frac{3}{7} x^{\frac{7}{3}} + \frac{4}{9} x^{\frac{9}{4}}]_{0}^{1}$

Step 7.2

Substitute and simplify.

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Step 7.2.1

Evaluate $\frac{3}{7} x^{\frac{7}{3}} + \frac{4}{9} x^{\frac{9}{4}}$ at $1$ and at $0$ .

$(\frac{3}{7} \cdot 1^{\frac{7}{3}} + \frac{4}{9} \cdot 1^{\frac{9}{4}}) - (\frac{3}{7} \cdot 0^{\frac{7}{3}} + \frac{4}{9} \cdot 0^{\frac{9}{4}})$

Step 7.2.2

Simplify.

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Step 7.2.2.1

One to any power is one.

$\frac{3}{7} \cdot 1 + \frac{4}{9} \cdot 1^{\frac{9}{4}} - (\frac{3}{7} \cdot 0^{\frac{7}{3}} + \frac{4}{9} \cdot 0^{\frac{9}{4}})$

Step 7.2.2.2

Multiply $\frac{3}{7}$ by $1$ .

$\frac{3}{7} + \frac{4}{9} \cdot 1^{\frac{9}{4}} - (\frac{3}{7} \cdot 0^{\frac{7}{3}} + \frac{4}{9} \cdot 0^{\frac{9}{4}})$

Step 7.2.2.3

One to any power is one.

$\frac{3}{7} + \frac{4}{9} \cdot 1 - (\frac{3}{7} \cdot 0^{\frac{7}{3}} + \frac{4}{9} \cdot 0^{\frac{9}{4}})$

Step 7.2.2.4

Multiply $\frac{4}{9}$ by $1$ .

$\frac{3}{7} + \frac{4}{9} - (\frac{3}{7} \cdot 0^{\frac{7}{3}} + \frac{4}{9} \cdot 0^{\frac{9}{4}})$

Step 7.2.2.5

To write $\frac{3}{7}$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$\frac{3}{7} \cdot \frac{9}{9} + \frac{4}{9} - (\frac{3}{7} \cdot 0^{\frac{7}{3}} + \frac{4}{9} \cdot 0^{\frac{9}{4}})$

Step 7.2.2.6

To write $\frac{4}{9}$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$\frac{3}{7} \cdot \frac{9}{9} + \frac{4}{9} \cdot \frac{7}{7} - (\frac{3}{7} \cdot 0^{\frac{7}{3}} + \frac{4}{9} \cdot 0^{\frac{9}{4}})$

Step 7.2.2.7

Write each expression with a common denominator of $63$ , by multiplying each by an appropriate factor of $1$ .

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Step 7.2.2.7.1

Multiply $\frac{3}{7}$ by $\frac{9}{9}$ .

$\frac{3 \cdot 9}{7 \cdot 9} + \frac{4}{9} \cdot \frac{7}{7} - (\frac{3}{7} \cdot 0^{\frac{7}{3}} + \frac{4}{9} \cdot 0^{\frac{9}{4}})$

Step 7.2.2.7.2

Multiply $7$ by $9$ .

$\frac{3 \cdot 9}{63} + \frac{4}{9} \cdot \frac{7}{7} - (\frac{3}{7} \cdot 0^{\frac{7}{3}} + \frac{4}{9} \cdot 0^{\frac{9}{4}})$

Step 7.2.2.7.3

Multiply $\frac{4}{9}$ by $\frac{7}{7}$ .

$\frac{3 \cdot 9}{63} + \frac{4 \cdot 7}{9 \cdot 7} - (\frac{3}{7} \cdot 0^{\frac{7}{3}} + \frac{4}{9} \cdot 0^{\frac{9}{4}})$

Step 7.2.2.7.4

Multiply $9$ by $7$ .

$\frac{3 \cdot 9}{63} + \frac{4 \cdot 7}{63} - (\frac{3}{7} \cdot 0^{\frac{7}{3}} + \frac{4}{9} \cdot 0^{\frac{9}{4}})$

Step 7.2.2.8

Combine the numerators over the common denominator.

$\frac{3 \cdot 9 + 4 \cdot 7}{63} - (\frac{3}{7} \cdot 0^{\frac{7}{3}} + \frac{4}{9} \cdot 0^{\frac{9}{4}})$

Step 7.2.2.9

Simplify the numerator.

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Step 7.2.2.9.1

Multiply $3$ by $9$ .

$\frac{27 + 4 \cdot 7}{63} - (\frac{3}{7} \cdot 0^{\frac{7}{3}} + \frac{4}{9} \cdot 0^{\frac{9}{4}})$

Step 7.2.2.9.2

Multiply $4$ by $7$ .

$\frac{27 + 28}{63} - (\frac{3}{7} \cdot 0^{\frac{7}{3}} + \frac{4}{9} \cdot 0^{\frac{9}{4}})$

Step 7.2.2.9.3

Add $27$ and $28$ .

$\frac{55}{63} - (\frac{3}{7} \cdot 0^{\frac{7}{3}} + \frac{4}{9} \cdot 0^{\frac{9}{4}})$

Step 7.2.2.10

Rewrite $0$ as $0^{3}$ .

$\frac{55}{63} - (\frac{3}{7} \cdot {(0^{3})}^{\frac{7}{3}} + \frac{4}{9} \cdot 0^{\frac{9}{4}})$

Step 7.2.2.11

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

$\frac{55}{63} - (\frac{3}{7} \cdot 0^{3 (\frac{7}{3})} + \frac{4}{9} \cdot 0^{\frac{9}{4}})$

Step 7.2.2.12

Cancel the common factor of $3$ .

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Step 7.2.2.12.1

Cancel the common factor.

$\frac{55}{63} - (\frac{3}{7} \cdot 0^{3 (\frac{7}{3})} + \frac{4}{9} \cdot 0^{\frac{9}{4}})$

Step 7.2.2.12.2

Rewrite the expression.

$\frac{55}{63} - (\frac{3}{7} \cdot 0^{7} + \frac{4}{9} \cdot 0^{\frac{9}{4}})$

Step 7.2.2.13

Raising $0$ to any positive power yields $0$ .

$\frac{55}{63} - (\frac{3}{7} \cdot 0 + \frac{4}{9} \cdot 0^{\frac{9}{4}})$

Step 7.2.2.14

Multiply $\frac{3}{7}$ by $0$ .

$\frac{55}{63} - (0 + \frac{4}{9} \cdot 0^{\frac{9}{4}})$

Step 7.2.2.15

Rewrite $0$ as $0^{4}$ .

$\frac{55}{63} - (0 + \frac{4}{9} \cdot {(0^{4})}^{\frac{9}{4}})$

Step 7.2.2.16

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

$\frac{55}{63} - (0 + \frac{4}{9} \cdot 0^{4 (\frac{9}{4})})$

Step 7.2.2.17

Cancel the common factor of $4$ .

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Step 7.2.2.17.1

Cancel the common factor.

$\frac{55}{63} - (0 + \frac{4}{9} \cdot 0^{4 (\frac{9}{4})})$

Step 7.2.2.17.2

Rewrite the expression.

$\frac{55}{63} - (0 + \frac{4}{9} \cdot 0^{9})$

Step 7.2.2.18

Raising $0$ to any positive power yields $0$ .

$\frac{55}{63} - (0 + \frac{4}{9} \cdot 0)$

Step 7.2.2.19

Multiply $\frac{4}{9}$ by $0$ .

$\frac{55}{63} - (0 + 0)$

Step 7.2.2.20

Add $0$ and $0$ .

$\frac{55}{63} - 0$

Step 7.2.2.21

Multiply $- 1$ by $0$ .

$\frac{55}{63} + 0$

Step 7.2.2.22

Add $\frac{55}{63}$ and $0$ .

$\frac{55}{63}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{55}{63}$

Decimal Form:

$0.\overline{873015}$

Step 9