Calculus Examples

$\int (5 x^{- 3} - 3 x^{5} + 4 x^{4}) d x$

Step 1

Remove parentheses.

$\int 5 x^{- 3} - 3 x^{5} + 4 x^{4} d x$

Step 2

Split the single integral into multiple integrals.

$\int 5 x^{- 3} d x + \int - 3 x^{5} d x + \int 4 x^{4} d x$

Step 3

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

$5 \int x^{- 3} d x + \int - 3 x^{5} d x + \int 4 x^{4} d x$

Step 4

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

$5 (- \frac{1}{2} x^{- 2} + C) + \int - 3 x^{5} d x + \int 4 x^{4} d x$

Step 5

Simplify.

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Step 5.1

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

$5 (- \frac{x^{- 2}}{2} + C) + \int - 3 x^{5} d x + \int 4 x^{4} d x$

Step 5.2

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

$5 (- \frac{1}{2 x^{2}} + C) + \int - 3 x^{5} d x + \int 4 x^{4} d x$

Step 6

Since $- 3$ is constant with respect to $x$ , move $- 3$ out of the integral.

$5 (- \frac{1}{2 x^{2}} + C) - 3 \int x^{5} d x + \int 4 x^{4} d x$

Step 7

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

$5 (- \frac{1}{2 x^{2}} + C) - 3 (\frac{1}{6} x^{6} + C) + \int 4 x^{4} d x$

Step 8

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

$5 (- \frac{1}{2 x^{2}} + C) - 3 (\frac{1}{6} x^{6} + C) + 4 \int x^{4} d x$

Step 9

By the Power Rule, the integral of $x^{4}$ with respect to $x$ is $\frac{1}{5} x^{5}$ .

$5 (- \frac{1}{2 x^{2}} + C) - 3 (\frac{1}{6} x^{6} + C) + 4 (\frac{1}{5} x^{5} + C)$

Step 10

Simplify.

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Step 10.1

Simplify.

$- \frac{5}{2 x^{2}} - \frac{x^{6}}{2} + 4 (\frac{1}{5} x^{5}) + C$

Step 10.2

Simplify.

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Step 10.2.1

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

$- \frac{5}{2 x^{2}} - \frac{x^{6}}{2} + 4 \frac{x^{5}}{5} + C$

Step 10.2.2

Combine $4$ and $\frac{x^{5}}{5}$ .

$- \frac{5}{2 x^{2}} - \frac{x^{6}}{2} + \frac{4 x^{5}}{5} + C$

Step 11

Reorder terms.

$- \frac{5}{2 x^{2}} - \frac{1}{2} x^{6} + \frac{4}{5} x^{5} + C$