Calculus Examples

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

Step 1

Remove parentheses.

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

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

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

Step 4

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

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

Step 5

Since $6$ is constant with respect to $x$ , move $6$ out of the integral.

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

Step 6

Simplify the expression.

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Step 6.1

Multiply $6$ by $- 1$ .

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

Step 6.2

Reorder $x^{2}$ and $1$ .

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

Step 6.3

Rewrite $1$ as $1^{2}$ .

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

Step 7

The integral of $\frac{1}{1^{2} + x^{2}}$ with respect to $x$ is $\arctan (x) + C$ .

$\ln (| x |) + C - 6 (\arctan (x) + C)$

Step 8

Simplify.

$\ln (| x |) - 6 \arctan (x) + C$