Calculus Examples

$2 x^{3} - \frac{2}{3} x^{2} + 5 x$

Step 1

Write $2 x^{3} - \frac{2}{3} x^{2} + 5 x$ as a function.

$f (x) = 2 x^{3} - \frac{2}{3} x^{2} + 5 x$

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.

$F (x) = \int 2 x^{3} - \frac{2}{3} x^{2} + 5 x d x$

Step 4

Split the single integral into multiple integrals.

$\int 2 x^{3} d x + \int - \frac{2}{3} x^{2} d x + \int 5 x d x$

Step 5

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

$2 \int x^{3} d x + \int - \frac{2}{3} x^{2} d x + \int 5 x d x$

Step 6

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

$2 (\frac{1}{4} x^{4} + C) + \int - \frac{2}{3} x^{2} d x + \int 5 x d x$

Step 7

Since $- \frac{2}{3}$ is constant with respect to $x$ , move $- \frac{2}{3}$ out of the integral.

$2 (\frac{1}{4} x^{4} + C) - \frac{2}{3} \int x^{2} d x + \int 5 x d x$

Step 8

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

$2 (\frac{1}{4} x^{4} + C) - \frac{2}{3} (\frac{1}{3} x^{3} + C) + \int 5 x d x$

Step 9

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

$2 (\frac{1}{4} x^{4} + C) - \frac{2}{3} (\frac{1}{3} x^{3} + C) + 5 \int x d x$

Step 10

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

$2 (\frac{1}{4} x^{4} + C) - \frac{2}{3} (\frac{1}{3} x^{3} + C) + 5 (\frac{1}{2} x^{2} + C)$

Step 11

Simplify.

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Step 11.1

Simplify.

$\frac{x^{4}}{2} - \frac{2 x^{3}}{9} + 5 (\frac{1}{2} x^{2}) + C$

Step 11.2

Simplify.

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Step 11.2.1

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

$\frac{x^{4}}{2} - \frac{2 x^{3}}{9} + 5 \frac{x^{2}}{2} + C$

Step 11.2.2

Combine $5$ and $\frac{x^{2}}{2}$ .

$\frac{x^{4}}{2} - \frac{2 x^{3}}{9} + \frac{5 x^{2}}{2} + C$

Step 12

Reorder terms.

$\frac{1}{2} x^{4} - \frac{2}{9} x^{3} + \frac{5}{2} x^{2} + C$

Step 13

The answer is the antiderivative of the function $f (x) = 2 x^{3} - \frac{2}{3} x^{2} + 5 x$ .

$F (x) =$ $\frac{1}{2} x^{4} - \frac{2}{9} x^{3} + \frac{5}{2} x^{2} + C$