Calculus Examples

$\int \frac{x}{6} + \frac{6}{x} d x$

Step 1

Split the single integral into multiple integrals.

$\int \frac{x}{6} d x + \int \frac{6}{x} d x$

Step 2

Since $\frac{1}{6}$ is constant with respect to $x$ , move $\frac{1}{6}$ out of the integral.

$\frac{1}{6} \int x d x + \int \frac{6}{x} d x$

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

Simplify.

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Step 6.1

Simplify.

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

Step 6.2

Simplify.

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Step 6.2.1

Multiply $\frac{1}{6}$ by $\frac{1}{2}$ .

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

Step 6.2.2

Multiply $6$ by $2$ .

$\frac{1}{12} x^{2} + 6 \ln (| x |) + C$