Calculus Examples

$\int 13 x^{12} - \frac{12}{x^{12}} d x$

Step 1

Split the single integral into multiple integrals.

$\int 13 x^{12} d x + \int - \frac{12}{x^{12}} d x$

Step 2

Since $13$ is constant with respect to $x$ , move $13$ out of the integral.

$13 \int x^{12} d x + \int - \frac{12}{x^{12}} d x$

Step 3

By the Power Rule, the integral of $x^{12}$ with respect to $x$ is $\frac{1}{13} x^{13}$ .

$13 (\frac{1}{13} x^{13} + C) + \int - \frac{12}{x^{12}} d x$

Step 4

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

$13 (\frac{1}{13} x^{13} + C) - \int \frac{12}{x^{12}} d x$

Step 5

Since $12$ is constant with respect to $x$ , move $12$ out of the integral.

$13 (\frac{1}{13} x^{13} + C) - (12 \int \frac{1}{x^{12}} d x)$

Step 6

Simplify the expression.

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Step 6.1

Simplify.

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Step 6.1.1

Combine $\frac{1}{13}$ and $x^{13}$ .

$13 (\frac{x^{13}}{13} + C) - (12 \int \frac{1}{x^{12}} d x)$

Step 6.1.2

Multiply $12$ by $- 1$ .

$13 (\frac{x^{13}}{13} + C) - 12 \int \frac{1}{x^{12}} d x$

Step 6.2

Apply basic rules of exponents.

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Step 6.2.1

Move $x^{12}$ out of the denominator by raising it to the $- 1$ power.

$13 (\frac{x^{13}}{13} + C) - 12 \int {(x^{12})}^{- 1} d x$

Step 6.2.2

Multiply the exponents in ${(x^{12})}^{- 1}$ .

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Step 6.2.2.1

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

$13 (\frac{x^{13}}{13} + C) - 12 \int x^{12 \cdot - 1} d x$

Step 6.2.2.2

Multiply $12$ by $- 1$ .

$13 (\frac{x^{13}}{13} + C) - 12 \int x^{- 12} d x$

Step 7

By the Power Rule, the integral of $x^{- 12}$ with respect to $x$ is $- \frac{1}{11} x^{- 11}$ .

$13 (\frac{x^{13}}{13} + C) - 12 (- \frac{1}{11} x^{- 11} + C)$

Step 8

Simplify.

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Step 8.1

Simplify.

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Step 8.1.1

Combine $x^{- 11}$ and $\frac{1}{11}$ .

$13 (\frac{x^{13}}{13} + C) - 12 (- \frac{x^{- 11}}{11} + C)$

Step 8.1.2

Move $x^{- 11}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$13 (\frac{x^{13}}{13} + C) - 12 (- \frac{1}{11 x^{11}} + C)$

Step 8.2

Simplify.

$x^{13} - 12 (- \frac{1}{11 x^{11}}) + C$

Step 8.3

Simplify.

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Step 8.3.1

Multiply $- 1$ by $- 12$ .

$x^{13} + 12 \frac{1}{11 x^{11}} + C$

Step 8.3.2

Combine $12$ and $\frac{1}{11 x^{11}}$ .

$x^{13} + \frac{12}{11 x^{11}} + C$