Calculus Examples

$\sqrt{x} + \frac{1}{x}$

Step 1

Write $\sqrt{x} + \frac{1}{x}$ as a function.

$f (x) = \sqrt{x} + \frac{1}{x}$

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.

$F (x) = \int \sqrt{x} + \frac{1}{x} d x$

Step 4

Split the single integral into multiple integrals.

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

Step 5

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

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

Step 6

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

$\frac{2}{3} x^{\frac{3}{2}} + C + \int \frac{1}{x} d x$

Step 7

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

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

Step 8

Simplify.

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

Step 9

The answer is the antiderivative of the function $f (x) = \sqrt{x} + \frac{1}{x}$ .

$F (x) =$ $\frac{2}{3} x^{\frac{3}{2}} + \ln (| x |) + C$