Calculus Examples

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

Step 1

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

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

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.

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

Step 4

Split the single integral into multiple integrals.

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

Step 5

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

$\frac{1}{4} x^{4} + C + \int \frac{x^{2}}{2} d x$

Step 6

Since $\frac{1}{2}$ is constant with respect to $x$ , move $\frac{1}{2}$ out of the integral.

$\frac{1}{4} x^{4} + C + \frac{1}{2} \int x^{2} d x$

Step 7

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

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

Step 8

Simplify.

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Step 8.1

Simplify.

$\frac{1}{4} x^{4} + \frac{1}{2} \cdot \frac{1}{3} x^{3} + C$

Step 8.2

Simplify.

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Step 8.2.1

Multiply $\frac{1}{2}$ by $\frac{1}{3}$ .

$\frac{1}{4} x^{4} + \frac{1}{2 \cdot 3} x^{3} + C$

Step 8.2.2

Multiply $2$ by $3$ .

$\frac{1}{4} x^{4} + \frac{1}{6} x^{3} + C$

Step 9

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

$F (x) =$ $\frac{1}{4} x^{4} + \frac{1}{6} x^{3} + C$