Calculus Examples

$\frac{5}{x} + \sqrt{x^{3}}$

Step 1

Write $\frac{5}{x} + \sqrt{x^{3}}$ as a function.

$f (x) = \frac{5}{x} + \sqrt{x^{3}}$

Step 2

The function $F (x)$ can be found by finding the indefinite integral of the derivative $f (x)$ .

$F (x) = \int f (x) d x$

Step 3

Set up the integral to solve.

$F (x) = \int \frac{5}{x} + \sqrt{x^{3}} d x$

Step 4

Split the single integral into multiple integrals.

$\int \frac{5}{x} d x + \int \sqrt{x^{3}} d x$

Step 5

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

$5 \int \frac{1}{x} d x + \int \sqrt{x^{3}} d x$

Step 6

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

$5 (\ln (| x |) + C) + \int \sqrt{x^{3}} d x$

Step 7

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{3}}$ as $x^{\frac{3}{2}}$ .

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

Step 8

By the Power Rule, the integral of $x^{\frac{3}{2}}$ with respect to $x$ is $\frac{2}{5} x^{\frac{5}{2}}$ .

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

Step 9

Simplify.

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

Step 10

The answer is the antiderivative of the function $f (x) = \frac{5}{x} + \sqrt{x^{3}}$ .

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