Calculus Examples

$\int \frac{5}{x} + \frac{x}{5} d x$

Step 1

Split the single integral into multiple integrals.

$\int \frac{5}{x} d x + \int \frac{x}{5} d x$

Step 2

Since $5$ is constant with respect to $x$ , move $5$ out of the integral.

$5 \int \frac{1}{x} d x + \int \frac{x}{5} d x$

Step 3

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

$5 (\ln (| x |) + C) + \int \frac{x}{5} d x$

Step 4

Since $\frac{1}{5}$ is constant with respect to $x$ , move $\frac{1}{5}$ out of the integral.

$5 (\ln (| x |) + C) + \frac{1}{5} \int x d x$

Step 5

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

$5 (\ln (| x |) + C) + \frac{1}{5} (\frac{1}{2} x^{2} + C)$

Step 6

Simplify.

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Step 6.1

Simplify.

$5 \ln (| x |) + \frac{1}{5} \cdot \frac{1}{2} x^{2} + C$

Step 6.2

Simplify.

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Step 6.2.1

Multiply $\frac{1}{5}$ by $\frac{1}{2}$ .

$5 \ln (| x |) + \frac{1}{5 \cdot 2} x^{2} + C$

Step 6.2.2

Multiply $5$ by $2$ .

$5 \ln (| x |) + \frac{1}{10} x^{2} + C$