Calculus Examples

$f (x) = 8 \sqrt{x} + 5 \cos (x)$

Step 1

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 2

Set up the integral to solve.

$F (x) = \int 8 \sqrt{x} + 5 \cos (x) d x$

Step 3

Split the single integral into multiple integrals.

$\int 8 \sqrt{x} d x + \int 5 \cos (x) d x$

Step 4

Since $8$ is constant with respect to $x$ , move $8$ out of the integral.

$8 \int \sqrt{x} d x + \int 5 \cos (x) d x$

Step 5

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

$8 \int x^{\frac{1}{2}} d x + \int 5 \cos (x) d x$

Step 6

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

$8 (\frac{2}{3} x^{\frac{3}{2}} + C) + \int 5 \cos (x) d x$

Step 7

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

$8 (\frac{2}{3} x^{\frac{3}{2}} + C) + 5 \int \cos (x) d x$

Step 8

The integral of $\cos (x)$ with respect to $x$ is $\sin (x)$ .

$8 (\frac{2}{3} x^{\frac{3}{2}} + C) + 5 (\sin (x) + C)$

Step 9

Simplify.

Tap for more steps...

Step 9.1

Simplify.

$8 (\frac{2}{3}) x^{\frac{3}{2}} + 5 \sin (x) + C$

Step 9.2

Simplify.

Tap for more steps...

Step 9.2.1

Combine $8$ and $\frac{2}{3}$ .

$\frac{8 \cdot 2}{3} x^{\frac{3}{2}} + 5 \sin (x) + C$

Step 9.2.2

Multiply $8$ by $2$ .

$\frac{16}{3} x^{\frac{3}{2}} + 5 \sin (x) + C$

Step 10

The answer is the antiderivative of the function $f (x) = 8 \sqrt{x} + 5 \cos (x)$ .

$F (x) =$ $\frac{16}{3} x^{\frac{3}{2}} + 5 \sin (x) + C$