Calculus Examples

$\int_{0}^{1} x (4 \sqrt[3]{x} + 5 \sqrt[4]{x}) d x$

Step 1

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

$\int_{0}^{1} x (4 x^{\frac{1}{3}} + 5 \sqrt[4]{x}) d x$

Step 2

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

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

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

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

Step 3.2

Reorder $x$ and $4$ .

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

Step 3.3

Reorder $x$ and $5$ .

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

Step 3.4

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

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

Step 3.5

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

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

Step 3.6

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

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

Step 3.7

Combine the numerators over the common denominator.

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

Step 3.8

Add $3$ and $1$ .

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

Step 3.9

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

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

Step 3.10

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

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

Step 3.11

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

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

Step 3.12

Combine the numerators over the common denominator.

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

Step 3.13

Add $4$ and $1$ .

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

Step 3.14

Reorder $4 x^{\frac{4}{3}}$ and $5 x^{\frac{5}{4}}$ .

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

Step 4

Split the single integral into multiple integrals.

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

Step 5

Since $5$ is constant with respect to $x$ , move $5$ out of the integral.

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

Step 6

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

$5 (\frac{4}{9} x^{\frac{9}{4}}]_{0}^{1}) + \int_{0}^{1} 4 x^{\frac{4}{3}} d x$

Step 7

Combine $\frac{4}{9}$ and $x^{\frac{9}{4}}$ .

$5 (\frac{4 x^{\frac{9}{4}}}{9}]_{0}^{1}) + \int_{0}^{1} 4 x^{\frac{4}{3}} d x$

Step 8

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

$5 (\frac{4 x^{\frac{9}{4}}}{9}]_{0}^{1}) + 4 \int_{0}^{1} x^{\frac{4}{3}} d x$

Step 9

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

$5 (\frac{4 x^{\frac{9}{4}}}{9}]_{0}^{1}) + 4 (\frac{3}{7} x^{\frac{7}{3}}]_{0}^{1})$

Step 10

Simplify the answer.

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Step 10.1

Combine $\frac{3}{7}$ and $x^{\frac{7}{3}}$ .

$5 (\frac{4 x^{\frac{9}{4}}}{9}]_{0}^{1}) + 4 (\frac{3 x^{\frac{7}{3}}}{7}]_{0}^{1})$

Step 10.2

Substitute and simplify.

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Step 10.2.1

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

$5 ((\frac{4 \cdot 1^{\frac{9}{4}}}{9}) - \frac{4 \cdot 0^{\frac{9}{4}}}{9}) + 4 (\frac{3 x^{\frac{7}{3}}}{7}]_{0}^{1})$

Step 10.2.2

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

$5 (\frac{4 \cdot 1^{\frac{9}{4}}}{9} - \frac{4 \cdot 0^{\frac{9}{4}}}{9}) + 4 (\frac{3 \cdot 1^{\frac{7}{3}}}{7} - \frac{3 \cdot 0^{\frac{7}{3}}}{7})$

Step 10.2.3

Simplify.

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Step 10.2.3.1

One to any power is one.

$5 (\frac{4 \cdot 1}{9} - \frac{4 \cdot 0^{\frac{9}{4}}}{9}) + 4 (\frac{3 \cdot 1^{\frac{7}{3}}}{7} - \frac{3 \cdot 0^{\frac{7}{3}}}{7})$

Step 10.2.3.2

Multiply $4$ by $1$ .

$5 (\frac{4}{9} - \frac{4 \cdot 0^{\frac{9}{4}}}{9}) + 4 (\frac{3 \cdot 1^{\frac{7}{3}}}{7} - \frac{3 \cdot 0^{\frac{7}{3}}}{7})$

Step 10.2.3.3

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

$5 (\frac{4}{9} - \frac{4 \cdot {(0^{4})}^{\frac{9}{4}}}{9}) + 4 (\frac{3 \cdot 1^{\frac{7}{3}}}{7} - \frac{3 \cdot 0^{\frac{7}{3}}}{7})$

Step 10.2.3.4

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

$5 (\frac{4}{9} - \frac{4 \cdot 0^{4 (\frac{9}{4})}}{9}) + 4 (\frac{3 \cdot 1^{\frac{7}{3}}}{7} - \frac{3 \cdot 0^{\frac{7}{3}}}{7})$

Step 10.2.3.5

Cancel the common factor of $4$ .

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Step 10.2.3.5.1

Cancel the common factor.

$5 (\frac{4}{9} - \frac{4 \cdot 0^{4 (\frac{9}{4})}}{9}) + 4 (\frac{3 \cdot 1^{\frac{7}{3}}}{7} - \frac{3 \cdot 0^{\frac{7}{3}}}{7})$

Step 10.2.3.5.2

Rewrite the expression.

$5 (\frac{4}{9} - \frac{4 \cdot 0^{9}}{9}) + 4 (\frac{3 \cdot 1^{\frac{7}{3}}}{7} - \frac{3 \cdot 0^{\frac{7}{3}}}{7})$

Step 10.2.3.6

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

$5 (\frac{4}{9} - \frac{4 \cdot 0}{9}) + 4 (\frac{3 \cdot 1^{\frac{7}{3}}}{7} - \frac{3 \cdot 0^{\frac{7}{3}}}{7})$

Step 10.2.3.7

Multiply $4$ by $0$ .

$5 (\frac{4}{9} - \frac{0}{9}) + 4 (\frac{3 \cdot 1^{\frac{7}{3}}}{7} - \frac{3 \cdot 0^{\frac{7}{3}}}{7})$

Step 10.2.3.8

Cancel the common factor of $0$ and $9$ .

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Step 10.2.3.8.1

Factor $9$ out of $0$ .

$5 (\frac{4}{9} - \frac{9 (0)}{9}) + 4 (\frac{3 \cdot 1^{\frac{7}{3}}}{7} - \frac{3 \cdot 0^{\frac{7}{3}}}{7})$

Step 10.2.3.8.2

Cancel the common factors.

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Step 10.2.3.8.2.1

Factor $9$ out of $9$ .

$5 (\frac{4}{9} - \frac{9 \cdot 0}{9 \cdot 1}) + 4 (\frac{3 \cdot 1^{\frac{7}{3}}}{7} - \frac{3 \cdot 0^{\frac{7}{3}}}{7})$

Step 10.2.3.8.2.2

Cancel the common factor.

$5 (\frac{4}{9} - \frac{9 \cdot 0}{9 \cdot 1}) + 4 (\frac{3 \cdot 1^{\frac{7}{3}}}{7} - \frac{3 \cdot 0^{\frac{7}{3}}}{7})$

Step 10.2.3.8.2.3

Rewrite the expression.

$5 (\frac{4}{9} - \frac{0}{1}) + 4 (\frac{3 \cdot 1^{\frac{7}{3}}}{7} - \frac{3 \cdot 0^{\frac{7}{3}}}{7})$

Step 10.2.3.8.2.4

Divide $0$ by $1$ .

$5 (\frac{4}{9} - 0) + 4 (\frac{3 \cdot 1^{\frac{7}{3}}}{7} - \frac{3 \cdot 0^{\frac{7}{3}}}{7})$

Step 10.2.3.9

Multiply $- 1$ by $0$ .

$5 (\frac{4}{9} + 0) + 4 (\frac{3 \cdot 1^{\frac{7}{3}}}{7} - \frac{3 \cdot 0^{\frac{7}{3}}}{7})$

Step 10.2.3.10

Add $\frac{4}{9}$ and $0$ .

$5 (\frac{4}{9}) + 4 (\frac{3 \cdot 1^{\frac{7}{3}}}{7} - \frac{3 \cdot 0^{\frac{7}{3}}}{7})$

Step 10.2.3.11

Combine $5$ and $\frac{4}{9}$ .

$\frac{5 \cdot 4}{9} + 4 (\frac{3 \cdot 1^{\frac{7}{3}}}{7} - \frac{3 \cdot 0^{\frac{7}{3}}}{7})$

Step 10.2.3.12

Multiply $5$ by $4$ .

$\frac{20}{9} + 4 (\frac{3 \cdot 1^{\frac{7}{3}}}{7} - \frac{3 \cdot 0^{\frac{7}{3}}}{7})$

Step 10.2.3.13

One to any power is one.

$\frac{20}{9} + 4 (\frac{3 \cdot 1}{7} - \frac{3 \cdot 0^{\frac{7}{3}}}{7})$

Step 10.2.3.14

Multiply $3$ by $1$ .

$\frac{20}{9} + 4 (\frac{3}{7} - \frac{3 \cdot 0^{\frac{7}{3}}}{7})$

Step 10.2.3.15

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

$\frac{20}{9} + 4 (\frac{3}{7} - \frac{3 \cdot {(0^{3})}^{\frac{7}{3}}}{7})$

Step 10.2.3.16

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

$\frac{20}{9} + 4 (\frac{3}{7} - \frac{3 \cdot 0^{3 (\frac{7}{3})}}{7})$

Step 10.2.3.17

Cancel the common factor of $3$ .

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Step 10.2.3.17.1

Cancel the common factor.

$\frac{20}{9} + 4 (\frac{3}{7} - \frac{3 \cdot 0^{3 (\frac{7}{3})}}{7})$

Step 10.2.3.17.2

Rewrite the expression.

$\frac{20}{9} + 4 (\frac{3}{7} - \frac{3 \cdot 0^{7}}{7})$

Step 10.2.3.18

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

$\frac{20}{9} + 4 (\frac{3}{7} - \frac{3 \cdot 0}{7})$

Step 10.2.3.19

Multiply $3$ by $0$ .

$\frac{20}{9} + 4 (\frac{3}{7} - \frac{0}{7})$

Step 10.2.3.20

Cancel the common factor of $0$ and $7$ .

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Step 10.2.3.20.1

Factor $7$ out of $0$ .

$\frac{20}{9} + 4 (\frac{3}{7} - \frac{7 (0)}{7})$

Step 10.2.3.20.2

Cancel the common factors.

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Step 10.2.3.20.2.1

Factor $7$ out of $7$ .

$\frac{20}{9} + 4 (\frac{3}{7} - \frac{7 \cdot 0}{7 \cdot 1})$

Step 10.2.3.20.2.2

Cancel the common factor.

$\frac{20}{9} + 4 (\frac{3}{7} - \frac{7 \cdot 0}{7 \cdot 1})$

Step 10.2.3.20.2.3

Rewrite the expression.

$\frac{20}{9} + 4 (\frac{3}{7} - \frac{0}{1})$

Step 10.2.3.20.2.4

Divide $0$ by $1$ .

$\frac{20}{9} + 4 (\frac{3}{7} - 0)$

Step 10.2.3.21

Multiply $- 1$ by $0$ .

$\frac{20}{9} + 4 (\frac{3}{7} + 0)$

Step 10.2.3.22

Add $\frac{3}{7}$ and $0$ .

$\frac{20}{9} + 4 (\frac{3}{7})$

Step 10.2.3.23

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

$\frac{20}{9} + \frac{4 \cdot 3}{7}$

Step 10.2.3.24

Multiply $4$ by $3$ .

$\frac{20}{9} + \frac{12}{7}$

Step 10.2.3.25

To write $\frac{20}{9}$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$\frac{20}{9} \cdot \frac{7}{7} + \frac{12}{7}$

Step 10.2.3.26

To write $\frac{12}{7}$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$\frac{20}{9} \cdot \frac{7}{7} + \frac{12}{7} \cdot \frac{9}{9}$

Step 10.2.3.27

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

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Step 10.2.3.27.1

Multiply $\frac{20}{9}$ by $\frac{7}{7}$ .

$\frac{20 \cdot 7}{9 \cdot 7} + \frac{12}{7} \cdot \frac{9}{9}$

Step 10.2.3.27.2

Multiply $9$ by $7$ .

$\frac{20 \cdot 7}{63} + \frac{12}{7} \cdot \frac{9}{9}$

Step 10.2.3.27.3

Multiply $\frac{12}{7}$ by $\frac{9}{9}$ .

$\frac{20 \cdot 7}{63} + \frac{12 \cdot 9}{7 \cdot 9}$

Step 10.2.3.27.4

Multiply $7$ by $9$ .

$\frac{20 \cdot 7}{63} + \frac{12 \cdot 9}{63}$

Step 10.2.3.28

Combine the numerators over the common denominator.

$\frac{20 \cdot 7 + 12 \cdot 9}{63}$

Step 10.2.3.29

Simplify the numerator.

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Step 10.2.3.29.1

Multiply $20$ by $7$ .

$\frac{140 + 12 \cdot 9}{63}$

Step 10.2.3.29.2

Multiply $12$ by $9$ .

$\frac{140 + 108}{63}$

Step 10.2.3.29.3

Add $140$ and $108$ .

$\frac{248}{63}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$\frac{248}{63}$

Decimal Form:

$3.\overline{936507}$

Mixed Number Form:

$3 \frac{59}{63}$

Step 12