Calculus Examples

$\int (- 5 x^{- 3} + 5 x^{5}) d x$

Step 1

Remove parentheses.

$\int - 5 x^{- 3} + 5 x^{5} d x$

Step 2

Split the single integral into multiple integrals.

$\int - 5 x^{- 3} d x + \int 5 x^{5} d x$

Step 3

Since $- 5$ is constant with respect to $x$ , move $- 5$ out of the integral.

$- 5 \int x^{- 3} d x + \int 5 x^{5} d x$

Step 4

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

$- 5 (- \frac{1}{2} x^{- 2} + C) + \int 5 x^{5} d x$

Step 5

Simplify.

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Step 5.1

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

$- 5 (- \frac{x^{- 2}}{2} + C) + \int 5 x^{5} d x$

Step 5.2

Move $x^{- 2}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- 5 (- \frac{1}{2 x^{2}} + C) + \int 5 x^{5} d x$

Step 6

Since $5$ is constant with respect to $x$ , move $5$ out of the integral.

$- 5 (- \frac{1}{2 x^{2}} + C) + 5 \int x^{5} d x$

Step 7

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

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

Step 8

Simplify.

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Step 8.1

Simplify.

$\frac{5}{2 x^{2}} + 5 (\frac{1}{6} x^{6}) + C$

Step 8.2

Simplify.

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Step 8.2.1

Combine $\frac{1}{6}$ and $x^{6}$ .

$\frac{5}{2 x^{2}} + 5 \frac{x^{6}}{6} + C$

Step 8.2.2

Combine $5$ and $\frac{x^{6}}{6}$ .

$\frac{5}{2 x^{2}} + \frac{5 x^{6}}{6} + C$

Step 8.3

Reorder terms.

$\frac{5}{2 x^{2}} + \frac{5}{6} x^{6} + C$