Calculus Examples

$\int 12 x^{11} - \frac{11}{x^{11}} d x$

Step 1

Split the single integral into multiple integrals.

$\int 12 x^{11} d x + \int - \frac{11}{x^{11}} d x$

Step 2

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

$12 \int x^{11} d x + \int - \frac{11}{x^{11}} d x$

Step 3

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

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

Step 4

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

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

Step 5

Since $11$ is constant with respect to $x$ , move $11$ out of the integral.

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

Step 6

Simplify the expression.

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Step 6.1

Simplify.

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Step 6.1.1

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

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

Step 6.1.2

Multiply $11$ by $- 1$ .

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

Step 6.2

Apply basic rules of exponents.

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Step 6.2.1

Move $x^{11}$ out of the denominator by raising it to the $- 1$ power.

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

Step 6.2.2

Multiply the exponents in ${(x^{11})}^{- 1}$ .

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Step 6.2.2.1

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

$12 (\frac{x^{12}}{12} + C) - 11 \int x^{11 \cdot - 1} d x$

Step 6.2.2.2

Multiply $11$ by $- 1$ .

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

Step 7

By the Power Rule, the integral of $x^{- 11}$ with respect to $x$ is $- \frac{1}{10} x^{- 10}$ .

$12 (\frac{x^{12}}{12} + C) - 11 (- \frac{1}{10} x^{- 10} + C)$

Step 8

Simplify.

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Step 8.1

Simplify.

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Step 8.1.1

Combine $x^{- 10}$ and $\frac{1}{10}$ .

$12 (\frac{x^{12}}{12} + C) - 11 (- \frac{x^{- 10}}{10} + C)$

Step 8.1.2

Move $x^{- 10}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$12 (\frac{x^{12}}{12} + C) - 11 (- \frac{1}{10 x^{10}} + C)$

Step 8.2

Simplify.

$x^{12} - 11 (- \frac{1}{10 x^{10}}) + C$

Step 8.3

Simplify.

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Step 8.3.1

Multiply $- 1$ by $- 11$ .

$x^{12} + 11 \frac{1}{10 x^{10}} + C$

Step 8.3.2

Combine $11$ and $\frac{1}{10 x^{10}}$ .

$x^{12} + \frac{11}{10 x^{10}} + C$