Calculus Examples

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

Step 1

Remove parentheses.

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

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

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

Step 4

By the Power Rule, the integral of $x^{3}$ with respect to $x$ is $\frac{1}{4} x^{4}$ .

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

Simplify.

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Step 9.1

Simplify.

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Step 9.1.1

Combine $\frac{1}{4}$ and $x^{4}$ .

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

Step 9.1.2

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

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

Step 9.1.3

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

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

Step 9.1.4

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

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

Step 9.2

Simplify.

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

Step 9.3

Simplify.

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Step 9.3.1

Multiply $- 1$ by $5$ .

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

Step 9.3.2

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

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

Step 9.3.3

Move the negative in front of the fraction.

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

Step 10

Reorder terms.

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