Calculus Examples

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

Step 1

Split the single integral into multiple integrals.

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

Step 2

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

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

Step 3

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

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

Step 4

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

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

Step 5

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

$\frac{1}{3} x^{3} + C - (\frac{1}{2} x^{2} + C) + \ln (| x |) + C$

Step 6

Simplify.

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Step 6.1

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

$\frac{1}{3} x^{3} + C - (\frac{x^{2}}{2} + C) + \ln (| x |) + C$

Step 6.2

Simplify.

$\frac{1}{3} x^{3} - \frac{1}{2} x^{2} + \ln (| x |) + C$