Calculus Examples

$\int (- 20 x^{3} - 6 x^{- 3}) d x$

Step 1

Remove parentheses.

$\int - 20 x^{3} - 6 x^{- 3} d x$

Step 2

Split the single integral into multiple integrals.

$\int - 20 x^{3} d x + \int - 6 x^{- 3} d x$

Step 3

Since $- 20$ is constant with respect to $x$ , move $- 20$ out of the integral.

$- 20 \int x^{3} d x + \int - 6 x^{- 3} d x$

Step 4

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

$- 20 (\frac{1}{4} x^{4} + C) + \int - 6 x^{- 3} d x$

Step 5

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

$- 20 (\frac{1}{4} x^{4} + C) - 6 \int x^{- 3} d x$

Step 6

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

$- 20 (\frac{1}{4} x^{4} + C) - 6 (- \frac{1}{2} x^{- 2} + C)$

Step 7

Simplify.

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Step 7.1

Simplify.

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Step 7.1.1

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

$- 20 (\frac{x^{4}}{4} + C) - 6 (- \frac{1}{2} x^{- 2} + C)$

Step 7.1.2

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

$- 20 (\frac{x^{4}}{4} + C) - 6 (- \frac{x^{- 2}}{2} + C)$

Step 7.1.3

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

$- 20 (\frac{x^{4}}{4} + C) - 6 (- \frac{1}{2 x^{2}} + C)$

Step 7.2

Simplify.

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

Step 7.3

Simplify.

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Step 7.3.1

Multiply $- 1$ by $- 6$ .

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

Step 7.3.2

Combine $6$ and $\frac{1}{2 x^{2}}$ .

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

Step 7.3.3

Cancel the common factor of $6$ and $2$ .

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Step 7.3.3.1

Factor $2$ out of $6$ .

$- 5 x^{4} + \frac{2 \cdot 3}{2 x^{2}} + C$

Step 7.3.3.2

Cancel the common factors.

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Step 7.3.3.2.1

Factor $2$ out of $2 x^{2}$ .

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

Step 7.3.3.2.2

Cancel the common factor.

$- 5 x^{4} + \frac{2 \cdot 3}{2 x^{2}} + C$

Step 7.3.3.2.3

Rewrite the expression.

$- 5 x^{4} + \frac{3}{x^{2}} + C$