Calculus Examples

$f (x) = 2 x + 4 x^{3} + \sqrt{x^{3}} + 8$

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.

$F (x) = \int 2 x + 4 x^{3} + \sqrt{x^{3}} + 8 d x$

Step 3

Split the single integral into multiple integrals.

$\int 2 x d x + \int 4 x^{3} d x + \int \sqrt{x^{3}} d x + \int 8 d x$

Step 4

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$2 \int x d x + \int 4 x^{3} d x + \int \sqrt{x^{3}} d x + \int 8 d x$

Step 5

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

$2 (\frac{1}{2} x^{2} + C) + \int 4 x^{3} d x + \int \sqrt{x^{3}} d x + \int 8 d x$

Step 6

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

$2 (\frac{1}{2} x^{2} + C) + 4 \int x^{3} d x + \int \sqrt{x^{3}} d x + \int 8 d x$

Step 7

By the Power Rule, the integral of $x^{3}$ with respect to $x$ is $\frac{1}{4} x^{4}$ .

$2 (\frac{1}{2} x^{2} + C) + 4 (\frac{1}{4} x^{4} + C) + \int \sqrt{x^{3}} d x + \int 8 d x$

Step 8

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{3}}$ as $x^{\frac{3}{2}}$ .

$2 (\frac{1}{2} x^{2} + C) + 4 (\frac{1}{4} x^{4} + C) + \int x^{\frac{3}{2}} d x + \int 8 d x$

Step 9

By the Power Rule, the integral of $x^{\frac{3}{2}}$ with respect to $x$ is $\frac{2}{5} x^{\frac{5}{2}}$ .

$2 (\frac{1}{2} x^{2} + C) + 4 (\frac{1}{4} x^{4} + C) + \frac{2}{5} x^{\frac{5}{2}} + C + \int 8 d x$

Step 10

Simplify.

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Step 10.1

Combine $\frac{1}{2}$ and $x^{2}$ .

$2 (\frac{x^{2}}{2} + C) + 4 (\frac{1}{4} x^{4} + C) + \frac{2}{5} x^{\frac{5}{2}} + C + \int 8 d x$

Step 10.2

Combine $\frac{1}{4}$ and $x^{4}$ .

$2 (\frac{x^{2}}{2} + C) + 4 (\frac{x^{4}}{4} + C) + \frac{2}{5} x^{\frac{5}{2}} + C + \int 8 d x$

Step 11

Apply the constant rule.

$2 (\frac{x^{2}}{2} + C) + 4 (\frac{x^{4}}{4} + C) + \frac{2}{5} x^{\frac{5}{2}} + C + 8 x + C$

Step 12

Simplify.

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Step 12.1

Simplify.

$x^{2} + x^{4} + \frac{2 x^{\frac{5}{2}}}{5} + 8 x + C$

Step 12.2

Reorder terms.

$x^{2} + x^{4} + \frac{2}{5} x^{\frac{5}{2}} + 8 x + C$

Step 13

The answer is the antiderivative of the function $f (x) = 2 x + 4 x^{3} + \sqrt{x^{3}} + 8$ .

$F (x) =$ $x^{2} + x^{4} + \frac{2}{5} x^{\frac{5}{2}} + 8 x + C$