Calculus Examples

$\int (12 x^{- 6} + \frac{17}{3 x}) d x$

Step 1

Remove parentheses.

$\int 12 x^{- 6} + \frac{17}{3 x} d x$

Step 2

Split the single integral into multiple integrals.

$\int 12 x^{- 6} d x + \int \frac{17}{3 x} d x$

Step 3

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

$12 \int x^{- 6} d x + \int \frac{17}{3 x} d x$

Step 4

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

$12 (- \frac{1}{5} x^{- 5} + C) + \int \frac{17}{3 x} d x$

Step 5

Simplify.

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Step 5.1

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

$12 (- \frac{x^{- 5}}{5} + C) + \int \frac{17}{3 x} d x$

Step 5.2

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

$12 (- \frac{1}{5 x^{5}} + C) + \int \frac{17}{3 x} d x$

Step 6

Since $\frac{17}{3}$ is constant with respect to $x$ , move $\frac{17}{3}$ out of the integral.

$12 (- \frac{1}{5 x^{5}} + C) + \frac{17}{3} \int \frac{1}{x} d x$

Step 7

The integral of $\frac{1}{x}$ with respect to $x$ is $\ln (| x |)$ .

$12 (- \frac{1}{5 x^{5}} + C) + \frac{17}{3} (\ln (| x |) + C)$

Step 8

Simplify.

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Step 8.1

Simplify.

$\frac{- 12}{5 x^{5}} + \frac{17}{3} \ln (| x |) + C$

Step 8.2

Move the negative in front of the fraction.

$- \frac{12}{5 x^{5}} + \frac{17}{3} \ln (| x |) + C$