Calculus Examples

$\int (\frac{16}{x^{6}} - \frac{9}{x^{- 3}} + \frac{4}{x^{\frac{1}{3}}}) d x$

Step 1

Remove parentheses.

$\int \frac{16}{x^{6}} - \frac{9}{x^{- 3}} + \frac{4}{x^{\frac{1}{3}}} d x$

Step 2

Simplify.

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Step 2.1

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

$\int \frac{16}{x^{6}} - (9 x^{3}) + \frac{4}{x^{\frac{1}{3}}} d x$

Step 2.2

Multiply $9$ by $- 1$ .

$\int \frac{16}{x^{6}} - 9 x^{3} + \frac{4}{x^{\frac{1}{3}}} d x$

Step 3

Split the single integral into multiple integrals.

$\int \frac{16}{x^{6}} d x + \int - 9 x^{3} d x + \int \frac{4}{x^{\frac{1}{3}}} d x$

Step 4

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

$16 \int \frac{1}{x^{6}} d x + \int - 9 x^{3} d x + \int \frac{4}{x^{\frac{1}{3}}} d x$

Step 5

Apply basic rules of exponents.

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Step 5.1

Move $x^{6}$ out of the denominator by raising it to the $- 1$ power.

$16 \int {(x^{6})}^{- 1} d x + \int - 9 x^{3} d x + \int \frac{4}{x^{\frac{1}{3}}} d x$

Step 5.2

Multiply the exponents in ${(x^{6})}^{- 1}$ .

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Step 5.2.1

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

$16 \int x^{6 \cdot - 1} d x + \int - 9 x^{3} d x + \int \frac{4}{x^{\frac{1}{3}}} d x$

Step 5.2.2

Multiply $6$ by $- 1$ .

$16 \int x^{- 6} d x + \int - 9 x^{3} d x + \int \frac{4}{x^{\frac{1}{3}}} d x$

Step 6

By the Power Rule, the integral of $x^{- 6}$ with respect to $x$ is $- \frac{1}{5} x^{- 5}$ .

$16 (- \frac{1}{5} x^{- 5} + C) + \int - 9 x^{3} d x + \int \frac{4}{x^{\frac{1}{3}}} d x$

Step 7

Simplify.

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Step 7.1

Combine $x^{- 5}$ and $\frac{1}{5}$ .

$16 (- \frac{x^{- 5}}{5} + C) + \int - 9 x^{3} d x + \int \frac{4}{x^{\frac{1}{3}}} d x$

Step 7.2

Move $x^{- 5}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$16 (- \frac{1}{5 x^{5}} + C) + \int - 9 x^{3} d x + \int \frac{4}{x^{\frac{1}{3}}} d x$

Step 8

Since $- 9$ is constant with respect to $x$ , move $- 9$ out of the integral.

$16 (- \frac{1}{5 x^{5}} + C) - 9 \int x^{3} d x + \int \frac{4}{x^{\frac{1}{3}}} d x$

Step 9

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

$16 (- \frac{1}{5 x^{5}} + C) - 9 (\frac{1}{4} x^{4} + C) + \int \frac{4}{x^{\frac{1}{3}}} d x$

Step 10

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

$16 (- \frac{1}{5 x^{5}} + C) - 9 (\frac{1}{4} x^{4} + C) + 4 \int \frac{1}{x^{\frac{1}{3}}} d x$

Step 11

Simplify the expression.

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Step 11.1

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

$16 (- \frac{1}{5 x^{5}} + C) - 9 (\frac{1}{4} x^{4} + C) + 4 \int {(x^{\frac{1}{3}})}^{- 1} d x$

Step 11.2

Simplify.

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Step 11.2.1

Combine $\frac{1}{4}$ and $x^{4}$ .

$16 (- \frac{1}{5 x^{5}} + C) - 9 (\frac{x^{4}}{4} + C) + 4 \int {(x^{\frac{1}{3}})}^{- 1} d x$

Step 11.2.2

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

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Step 11.2.2.1

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

$16 (- \frac{1}{5 x^{5}} + C) - 9 (\frac{x^{4}}{4} + C) + 4 \int x^{\frac{1}{3} \cdot - 1} d x$

Step 11.2.2.2

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

$16 (- \frac{1}{5 x^{5}} + C) - 9 (\frac{x^{4}}{4} + C) + 4 \int x^{\frac{- 1}{3}} d x$

Step 11.2.2.3

Move the negative in front of the fraction.

$16 (- \frac{1}{5 x^{5}} + C) - 9 (\frac{x^{4}}{4} + C) + 4 \int x^{- \frac{1}{3}} d x$

Step 12

By the Power Rule, the integral of $x^{- \frac{1}{3}}$ with respect to $x$ is $\frac{3}{2} x^{\frac{2}{3}}$ .

$16 (- \frac{1}{5 x^{5}} + C) - 9 (\frac{x^{4}}{4} + C) + 4 (\frac{3}{2} x^{\frac{2}{3}} + C)$

Step 13

Simplify.

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Step 13.1

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

$16 (- \frac{1}{5 x^{5}} + C) - 9 (\frac{x^{4}}{4} + C) + 4 (\frac{3 x^{\frac{2}{3}}}{2} + C)$

Step 13.2

Simplify.

$- \frac{16}{5 x^{5}} - \frac{9 x^{4}}{4} + 4 \frac{3 x^{\frac{2}{3}}}{2} + C$

Step 13.3

Simplify.

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Step 13.3.1

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

$- \frac{16}{5 x^{5}} - \frac{9 x^{4}}{4} + \frac{4 (3 x^{\frac{2}{3}})}{2} + C$

Step 13.3.2

Multiply $3$ by $4$ .

$- \frac{16}{5 x^{5}} - \frac{9 x^{4}}{4} + \frac{12 x^{\frac{2}{3}}}{2} + C$

Step 13.3.3

Factor $2$ out of $12 x^{\frac{2}{3}}$ .

$- \frac{16}{5 x^{5}} - \frac{9 x^{4}}{4} + \frac{2 (6 x^{\frac{2}{3}})}{2} + C$

Step 13.3.4

Cancel the common factors.

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Step 13.3.4.1

Factor $2$ out of $2$ .

$- \frac{16}{5 x^{5}} - \frac{9 x^{4}}{4} + \frac{2 (6 x^{\frac{2}{3}})}{2 (1)} + C$

Step 13.3.4.2

Cancel the common factor.

$- \frac{16}{5 x^{5}} - \frac{9 x^{4}}{4} + \frac{2 (6 x^{\frac{2}{3}})}{2 \cdot 1} + C$

Step 13.3.4.3

Rewrite the expression.

$- \frac{16}{5 x^{5}} - \frac{9 x^{4}}{4} + \frac{6 x^{\frac{2}{3}}}{1} + C$

Step 13.3.4.4

Divide $6 x^{\frac{2}{3}}$ by $1$ .

$- \frac{16}{5 x^{5}} - \frac{9 x^{4}}{4} + 6 x^{\frac{2}{3}} + C$

Step 14

Reorder terms.

$- \frac{16}{5 x^{5}} - \frac{9}{4} x^{4} + 6 x^{\frac{2}{3}} + C$