Calculus Examples

$\int 16 \sqrt[3]{x^{5}} - \frac{16}{\sqrt[3]{x^{5}}} d x$

Step 1

Split the single integral into multiple integrals.

$\int 16 \sqrt[3]{x^{5}} d x + \int - \frac{16}{\sqrt[3]{x^{5}}} d x$

Step 2

Since $16$ is constant with respect to $x$ , move $16$ out of the integral.

$16 \int \sqrt[3]{x^{5}} d x + \int - \frac{16}{\sqrt[3]{x^{5}}} d x$

Step 3

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x^{5}}$ as $x^{\frac{5}{3}}$ .

$16 \int x^{\frac{5}{3}} d x + \int - \frac{16}{\sqrt[3]{x^{5}}} d x$

Step 4

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

$16 (\frac{3}{8} x^{\frac{8}{3}} + C) + \int - \frac{16}{\sqrt[3]{x^{5}}} d x$

Step 5

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$16 (\frac{3}{8} x^{\frac{8}{3}} + C) - \int \frac{16}{\sqrt[3]{x^{5}}} d x$

Step 6

Since $16$ is constant with respect to $x$ , move $16$ out of the integral.

$16 (\frac{3}{8} x^{\frac{8}{3}} + C) - (16 \int \frac{1}{\sqrt[3]{x^{5}}} d x)$

Step 7

Simplify the expression.

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Step 7.1

Simplify.

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Step 7.1.1

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

$16 (\frac{3 x^{\frac{8}{3}}}{8} + C) - (16 \int \frac{1}{\sqrt[3]{x^{5}}} d x)$

Step 7.1.2

Multiply $16$ by $- 1$ .

$16 (\frac{3 x^{\frac{8}{3}}}{8} + C) - 16 \int \frac{1}{\sqrt[3]{x^{5}}} d x$

Step 7.2

Apply basic rules of exponents.

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Step 7.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x^{5}}$ as $x^{\frac{5}{3}}$ .

$16 (\frac{3 x^{\frac{8}{3}}}{8} + C) - 16 \int \frac{1}{x^{\frac{5}{3}}} d x$

Step 7.2.2

Move $x^{\frac{5}{3}}$ out of the denominator by raising it to the $- 1$ power.

$16 (\frac{3 x^{\frac{8}{3}}}{8} + C) - 16 \int {(x^{\frac{5}{3}})}^{- 1} d x$

Step 7.2.3

Multiply the exponents in ${(x^{\frac{5}{3}})}^{- 1}$ .

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Step 7.2.3.1

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

$16 (\frac{3 x^{\frac{8}{3}}}{8} + C) - 16 \int x^{\frac{5}{3} \cdot - 1} d x$

Step 7.2.3.2

Multiply $\frac{5}{3} \cdot - 1$ .

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Step 7.2.3.2.1

Combine $\frac{5}{3}$ and $- 1$ .

$16 (\frac{3 x^{\frac{8}{3}}}{8} + C) - 16 \int x^{\frac{5 \cdot - 1}{3}} d x$

Step 7.2.3.2.2

Multiply $5$ by $- 1$ .

$16 (\frac{3 x^{\frac{8}{3}}}{8} + C) - 16 \int x^{\frac{- 5}{3}} d x$

Step 7.2.3.3

Move the negative in front of the fraction.

$16 (\frac{3 x^{\frac{8}{3}}}{8} + C) - 16 \int x^{- \frac{5}{3}} d x$

Step 8

By the Power Rule, the integral of $x^{- \frac{5}{3}}$ with respect to $x$ is $- \frac{3}{2} x^{- \frac{2}{3}}$ .

$16 (\frac{3 x^{\frac{8}{3}}}{8} + C) - 16 (- \frac{3}{2} x^{- \frac{2}{3}} + C)$

Step 9

Simplify.

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Step 9.1

Simplify.

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Step 9.1.1

Combine $x^{- \frac{2}{3}}$ and $\frac{3}{2}$ .

$16 (\frac{3 x^{\frac{8}{3}}}{8} + C) - 16 (- \frac{x^{- \frac{2}{3}} \cdot 3}{2} + C)$

Step 9.1.2

Move $3$ to the left of $x^{- \frac{2}{3}}$ .

$16 (\frac{3 x^{\frac{8}{3}}}{8} + C) - 16 (- \frac{3 \cdot x^{- \frac{2}{3}}}{2} + C)$

Step 9.1.3

Move $x^{- \frac{2}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$16 (\frac{3 x^{\frac{8}{3}}}{8} + C) - 16 (- \frac{3}{2 x^{\frac{2}{3}}} + C)$

Step 9.2

Simplify.

$6 x^{\frac{8}{3}} - 16 (- \frac{3}{2 x^{\frac{2}{3}}}) + C$

Step 9.3

Simplify.

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Step 9.3.1

Multiply $- 1$ by $- 16$ .

$6 x^{\frac{8}{3}} + 16 \frac{3}{2 x^{\frac{2}{3}}} + C$

Step 9.3.2

Combine $16$ and $\frac{3}{2 x^{\frac{2}{3}}}$ .

$6 x^{\frac{8}{3}} + \frac{16 \cdot 3}{2 x^{\frac{2}{3}}} + C$

Step 9.3.3

Multiply $16$ by $3$ .

$6 x^{\frac{8}{3}} + \frac{48}{2 x^{\frac{2}{3}}} + C$

Step 9.3.4

Factor $2$ out of $48$ .

$6 x^{\frac{8}{3}} + \frac{2 \cdot 24}{2 x^{\frac{2}{3}}} + C$

Step 9.3.5

Cancel the common factors.

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Step 9.3.5.1

Factor $2$ out of $2 x^{\frac{2}{3}}$ .

$6 x^{\frac{8}{3}} + \frac{2 \cdot 24}{2 (x^{\frac{2}{3}})} + C$

Step 9.3.5.2

Cancel the common factor.

$6 x^{\frac{8}{3}} + \frac{2 \cdot 24}{2 x^{\frac{2}{3}}} + C$

Step 9.3.5.3

Rewrite the expression.

$6 x^{\frac{8}{3}} + \frac{24}{x^{\frac{2}{3}}} + C$