Calculus Examples

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

Step 1

Remove parentheses.

$\int 2 x^{4} - 6 x^{5} + 3 x^{- 3} d x$

Step 2

Split the single integral into multiple integrals.

$\int 2 x^{4} d x + \int - 6 x^{5} d x + \int 3 x^{- 3} d x$

Step 3

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

$2 \int x^{4} d x + \int - 6 x^{5} d x + \int 3 x^{- 3} d x$

Step 4

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

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

Step 5

Since $- 6$ is constant with respect to $x$ , move $- 6$ out of the integral.

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

Simplify.

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Step 9.1

Simplify.

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Step 9.1.1

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

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

Step 9.1.2

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

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

Step 9.1.3

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

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

Step 9.1.4

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

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

Step 9.2

Simplify.

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

Step 9.3

Simplify.

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Step 9.3.1

Multiply $- 1$ by $3$ .

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

Step 9.3.2

Combine $- 3$ and $\frac{1}{2 x^{2}}$ .

$\frac{2 x^{5}}{5} - x^{6} + \frac{- 3}{2 x^{2}} + C$

Step 9.3.3

Move the negative in front of the fraction.

$\frac{2 x^{5}}{5} - x^{6} - \frac{3}{2 x^{2}} + C$

Step 9.4

Reorder terms.

$\frac{2}{5} x^{5} - x^{6} - \frac{3}{2 x^{2}} + C$