Calculus Examples

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

Step 1

Remove parentheses.

$\int 2 x^{4} + 2 x^{2} - 5 x^{- 3} d x$

Step 2

Split the single integral into multiple integrals.

$\int 2 x^{4} d x + \int 2 x^{2} d x + \int - 5 x^{- 3} d x$

Step 3

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$2 \int x^{4} d x + \int 2 x^{2} d x + \int - 5 x^{- 3} d x$

Step 4

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

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

Step 5

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

Simplify.

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Step 9.1

Simplify.

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Step 9.1.1

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

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

Step 9.1.2

Combine $\frac{1}{3}$ and $x^{3}$ .

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

Step 9.1.3

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

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

Step 9.1.4

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

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

Step 9.2

Simplify.

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

Step 9.3

Simplify.

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Step 9.3.1

Multiply $- 1$ by $- 5$ .

$\frac{2 x^{5}}{5} + \frac{2 x^{3}}{3} + 5 \frac{1}{2 x^{2}} + C$

Step 9.3.2

Combine $5$ and $\frac{1}{2 x^{2}}$ .

$\frac{2 x^{5}}{5} + \frac{2 x^{3}}{3} + \frac{5}{2 x^{2}} + C$

Step 9.4

Reorder terms.

$\frac{2}{5} x^{5} + \frac{2}{3} x^{3} + \frac{5}{2 x^{2}} + C$