Calculus Examples

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

Step 1

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

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

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.

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

Step 4

Split the single integral into multiple integrals.

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

Step 5

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

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

Step 6

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

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

Step 7

Combine $\frac{1}{3}$ and $x^{3}$ .

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

Step 8

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

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

Step 9

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

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

Step 10

Simplify.

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Step 10.1

Simplify.

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

Step 10.2

Simplify.

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Step 10.2.1

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

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

Step 10.2.2

Multiply $\frac{1}{2}$ by $\frac{x^{2}}{2}$ .

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

Step 10.2.3

Multiply $2$ by $2$ .

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

Step 10.3

Reorder terms.

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

Step 11

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

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