Calculus Examples

$x^{\sqrt{31}}$

Step 1

Write $x^{\sqrt{31}}$ as a function.

$f (x) = x^{\sqrt{31}}$

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.

$F (x) = \int x^{\sqrt{31}} d x$

Step 4

By the Power Rule, the integral of $x^{\sqrt{31}}$ with respect to $x$ is $\frac{1}{\sqrt{31} + 1} x^{\sqrt{31} + 1}$ .

$\frac{1}{\sqrt{31} + 1} x^{\sqrt{31} + 1} + C$

Step 5

Simplify the answer.

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Step 5.1

Rewrite $\frac{1}{\sqrt{31} + 1} x^{\sqrt{31} + 1} + C$ as $\frac{\sqrt{31} - 1}{30} x^{\sqrt{31} + 1} + C$ .

$\frac{\sqrt{31} - 1}{30} x^{\sqrt{31} + 1} + C$

Step 5.2

Combine $\frac{\sqrt{31} - 1}{30}$ and $x^{\sqrt{31} + 1}$ .

$\frac{(\sqrt{31} - 1) x^{\sqrt{31} + 1}}{30} + C$

Step 6

The answer is the antiderivative of the function $f (x) = x^{\sqrt{31}}$ .

$F (x) =$ $\frac{(\sqrt{31} - 1) x^{\sqrt{31} + 1}}{30} + C$