Calculus Examples

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

Step 1

Remove parentheses.

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

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

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

Step 4

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

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

Step 5

The integral of $x^{- 1}$ with respect to $x$ is $\ln (| x |)$ .

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

Step 6

Simplify.

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