Calculus Examples

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

Step 1

Remove parentheses.

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

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

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

Step 4

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

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

Step 5

Apply the constant rule.

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

Simplify.

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Step 9.1

Simplify.

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Step 9.1.1

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

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

Step 9.1.2

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

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

Step 9.2

Simplify.

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

Step 9.3

Simplify.

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Step 9.3.1

Multiply $- 1$ by $2$ .

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

Step 9.3.2

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

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

Step 9.3.3

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

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Step 9.3.3.1

Factor $2$ out of $- 2$ .

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

Step 9.3.3.2

Cancel the common factors.

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Step 9.3.3.2.1

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

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

Step 9.3.3.2.2

Cancel the common factor.

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

Step 9.3.3.2.3

Rewrite the expression.

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

Step 9.3.4

Move the negative in front of the fraction.

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

Step 10

Reorder terms.

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