Calculus Examples

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

Step 1

Split the single integral into multiple integrals.

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

Step 2

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

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

Step 3

Apply basic rules of exponents.

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Step 3.1

Move $x^{2}$ out of the denominator by raising it to the $- 1$ power.

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

Step 3.2

Multiply the exponents in ${(x^{2})}^{- 1}$ .

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Step 3.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\ln (| x |) + C + \int x^{2 \cdot - 1} d x$

Step 3.2.2

Multiply $2$ by $- 1$ .

$\ln (| x |) + C + \int x^{- 2} d x$

Step 4

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

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

Step 5

Simplify.

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