Calculus Examples

$f (x) = 3 x^{2}$

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.

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

Step 3

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

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

Step 4

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

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

Step 5

Simplify the answer.

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Step 5.1

Rewrite $3 (\frac{1}{3} x^{3} + C)$ as $3 (\frac{1}{3}) x^{3} + C$ .

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

Step 5.2

Simplify.

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Step 5.2.1

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

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

Step 5.2.2

Cancel the common factor of $3$ .

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Step 5.2.2.1

Cancel the common factor.

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

Step 5.2.2.2

Rewrite the expression.

$1 x^{3} + C$

Step 5.2.3

Multiply $x^{3}$ by $1$ .

$x^{3} + C$

Step 6

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

$F (x) =$ $x^{3} + C$