Calculus Examples

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

Step 1

Write $x^{\frac{1}{2}} + 2$ as a function.

$f (x) = x^{\frac{1}{2}} + 2$

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.

$F (x) = \int x^{\frac{1}{2}} + 2 d x$

Step 4

Split the single integral into multiple integrals.

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

Step 5

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

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

Step 6

Apply the constant rule.

$\frac{2}{3} x^{\frac{3}{2}} + C + 2 x + C$

Step 7

Simplify.

$\frac{2}{3} x^{\frac{3}{2}} + 2 x + C$

Step 8

The answer is the antiderivative of the function $f (x) = x^{\frac{1}{2}} + 2$ .

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