Calculus Examples

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

Step 1

Remove parentheses.

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

Apply the constant rule.

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

Simplify.

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Step 7.1

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

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

Step 7.2

Simplify.

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

Step 7.3

Reorder terms.

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