Calculus Examples

$\int \sqrt{x} + \frac{4}{x} - 5 e^{x} d x$

Step 1

Split the single integral into multiple integrals.

$\int \sqrt{x} d x + \int \frac{4}{x} d x + \int - 5 e^{x} d x$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$\int x^{\frac{1}{2}} d x + \int \frac{4}{x} d x + \int - 5 e^{x} d x$

Step 3

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

$\frac{2}{3} x^{\frac{3}{2}} + C + \int \frac{4}{x} d x + \int - 5 e^{x} d x$

Step 4

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

$\frac{2}{3} x^{\frac{3}{2}} + C + 4 \int \frac{1}{x} d x + \int - 5 e^{x} d x$

Step 5

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

$\frac{2}{3} x^{\frac{3}{2}} + C + 4 (\ln (| x |) + C) + \int - 5 e^{x} d x$

Step 6

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

$\frac{2}{3} x^{\frac{3}{2}} + C + 4 (\ln (| x |) + C) - 5 \int e^{x} d x$

Step 7

The integral of $e^{x}$ with respect to $x$ is $e^{x}$ .

$\frac{2}{3} x^{\frac{3}{2}} + C + 4 (\ln (| x |) + C) - 5 (e^{x} + C)$

Step 8

Simplify.

$\frac{2}{3} x^{\frac{3}{2}} + 4 \ln (| x |) - 5 e^{x} + C$