Calculus Examples

$f (x) = 7 - 6 x^{2} + 12 x^{3}$

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.

$F (x) = \int 7 - 6 x^{2} + 12 x^{3} d x$

Step 3

Split the single integral into multiple integrals.

$\int 7 d x + \int - 6 x^{2} d x + \int 12 x^{3} d x$

Step 4

Apply the constant rule.

$7 x + C + \int - 6 x^{2} d x + \int 12 x^{3} d x$

Step 5

Since $- 6$ is constant with respect to $x$ , move $- 6$ out of the integral.

$7 x + C - 6 \int x^{2} d x + \int 12 x^{3} d x$

Step 6

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

$7 x + C - 6 (\frac{1}{3} x^{3} + C) + \int 12 x^{3} d x$

Step 7

Since $12$ is constant with respect to $x$ , move $12$ out of the integral.

$7 x + C - 6 (\frac{1}{3} x^{3} + C) + 12 \int x^{3} d x$

Step 8

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

$7 x + C - 6 (\frac{1}{3} x^{3} + C) + 12 (\frac{1}{4} x^{4} + C)$

Step 9

Simplify.

Tap for more steps...

Step 9.1

Simplify.

$7 x - 2 x^{3} + 12 (\frac{1}{4} x^{4}) + C$

Step 9.2

Simplify.

Tap for more steps...

Step 9.2.1

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

$7 x - 2 x^{3} + 12 \frac{x^{4}}{4} + C$

Step 9.2.2

Combine $12$ and $\frac{x^{4}}{4}$ .

$7 x - 2 x^{3} + \frac{12 x^{4}}{4} + C$

Step 9.2.3

Cancel the common factor of $12$ and $4$ .

Tap for more steps...

Step 9.2.3.1

Factor $4$ out of $12 x^{4}$ .

$7 x - 2 x^{3} + \frac{4 (3 x^{4})}{4} + C$

Step 9.2.3.2

Cancel the common factors.

Tap for more steps...

Step 9.2.3.2.1

Factor $4$ out of $4$ .

$7 x - 2 x^{3} + \frac{4 (3 x^{4})}{4 (1)} + C$

Step 9.2.3.2.2

Cancel the common factor.

$7 x - 2 x^{3} + \frac{4 (3 x^{4})}{4 \cdot 1} + C$

Step 9.2.3.2.3

Rewrite the expression.

$7 x - 2 x^{3} + \frac{3 x^{4}}{1} + C$

Step 9.2.3.2.4

Divide $3 x^{4}$ by $1$ .

$7 x - 2 x^{3} + 3 x^{4} + C$

Step 10

The answer is the antiderivative of the function $f (x) = 7 - 6 x^{2} + 12 x^{3}$ .

$F (x) =$ $7 x - 2 x^{3} + 3 x^{4} + C$