Calculus Examples

${(\sqrt{x})}^{11}$

Step 1

Write ${(\sqrt{x})}^{11}$ as a function.

$f (x) = {(\sqrt{x})}^{11}$

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.

$F (x) = \int {(\sqrt{x})}^{11} d x$

Step 4

Rewrite ${(\sqrt{x})}^{11}$ as $\sqrt{x^{11}}$ .

$\int \sqrt{x^{11}} d x$

Step 5

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{11}}$ as $x^{\frac{11}{2}}$ .

$\int x^{\frac{11}{2}} d x$

Step 6

By the Power Rule, the integral of $x^{\frac{11}{2}}$ with respect to $x$ is $\frac{2}{13} x^{\frac{13}{2}}$ .

$\frac{2}{13} x^{\frac{13}{2}} + C$

Step 7

The answer is the antiderivative of the function $f (x) = {(\sqrt{x})}^{11}$ .

$F (x) =$ $\frac{2}{13} x^{\frac{13}{2}} + C$