Calculus Examples

$\int (x^{- 7} + x^{- \frac{1}{3}} + x^{- 2}) d x$

Step 1

Remove parentheses.

$\int x^{- 7} + x^{- \frac{1}{3}} + x^{- 2} d x$

Step 2

Split the single integral into multiple integrals.

$\int x^{- 7} d x + \int x^{- \frac{1}{3}} d x + \int x^{- 2} d x$

Step 3

By the Power Rule, the integral of $x^{- 7}$ with respect to $x$ is $- \frac{1}{6} x^{- 6}$ .

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

Step 4

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

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

Step 5

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

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

Step 6

Simplify.

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Step 6.1

Simplify.

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

Step 6.2

Reorder terms.

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