Calculus Examples

$\int 9 \sqrt{x} - \frac{2}{x^{8}} d x$

Step 1

Split the single integral into multiple integrals.

$\int 9 \sqrt{x} d x + \int - \frac{2}{x^{8}} d x$

Step 2

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

$9 \int \sqrt{x} d x + \int - \frac{2}{x^{8}} d x$

Step 3

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$9 \int x^{\frac{1}{2}} d x + \int - \frac{2}{x^{8}} d x$

Step 4

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

$9 (\frac{2}{3} x^{\frac{3}{2}} + C) + \int - \frac{2}{x^{8}} d x$

Step 5

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

$9 (\frac{2}{3} x^{\frac{3}{2}} + C) - \int \frac{2}{x^{8}} d x$

Step 6

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$9 (\frac{2}{3} x^{\frac{3}{2}} + C) - (2 \int \frac{1}{x^{8}} d x)$

Step 7

Simplify the expression.

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Step 7.1

Simplify.

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Step 7.1.1

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

$9 (\frac{2 x^{\frac{3}{2}}}{3} + C) - (2 \int \frac{1}{x^{8}} d x)$

Step 7.1.2

Multiply $2$ by $- 1$ .

$9 (\frac{2 x^{\frac{3}{2}}}{3} + C) - 2 \int \frac{1}{x^{8}} d x$

Step 7.2

Apply basic rules of exponents.

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Step 7.2.1

Move $x^{8}$ out of the denominator by raising it to the $- 1$ power.

$9 (\frac{2 x^{\frac{3}{2}}}{3} + C) - 2 \int {(x^{8})}^{- 1} d x$

Step 7.2.2

Multiply the exponents in ${(x^{8})}^{- 1}$ .

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Step 7.2.2.1

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

$9 (\frac{2 x^{\frac{3}{2}}}{3} + C) - 2 \int x^{8 \cdot - 1} d x$

Step 7.2.2.2

Multiply $8$ by $- 1$ .

$9 (\frac{2 x^{\frac{3}{2}}}{3} + C) - 2 \int x^{- 8} d x$

Step 8

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

$9 (\frac{2 x^{\frac{3}{2}}}{3} + C) - 2 (- \frac{1}{7} x^{- 7} + C)$

Step 9

Simplify.

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Step 9.1

Simplify.

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Step 9.1.1

Combine $x^{- 7}$ and $\frac{1}{7}$ .

$9 (\frac{2 x^{\frac{3}{2}}}{3} + C) - 2 (- \frac{x^{- 7}}{7} + C)$

Step 9.1.2

Move $x^{- 7}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$9 (\frac{2 x^{\frac{3}{2}}}{3} + C) - 2 (- \frac{1}{7 x^{7}} + C)$

Step 9.2

Simplify.

$6 x^{\frac{3}{2}} - 2 (- \frac{1}{7 x^{7}}) + C$

Step 9.3

Simplify.

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Step 9.3.1

Multiply $- 1$ by $- 2$ .

$6 x^{\frac{3}{2}} + 2 \frac{1}{7 x^{7}} + C$

Step 9.3.2

Combine $2$ and $\frac{1}{7 x^{7}}$ .

$6 x^{\frac{3}{2}} + \frac{2}{7 x^{7}} + C$