Calculus Examples

$f (x) = x^{3} - 4 x^{2} + 4 x$

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.

$F (x) = \int x^{3} - 4 x^{2} + 4 x d x$

Step 3

Split the single integral into multiple integrals.

$\int x^{3} d x + \int - 4 x^{2} d x + \int 4 x d x$

Step 4

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

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

Step 5

Since $- 4$ is constant with respect to $x$ , move $- 4$ out of the integral.

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

Step 6

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

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

Step 7

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

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

Step 8

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

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

Step 9

Simplify.

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Step 9.1

Simplify.

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

Step 9.2

Simplify.

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Step 9.2.1

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

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

Step 9.2.2

Combine $4$ and $\frac{x^{2}}{2}$ .

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

Step 9.2.3

Cancel the common factor of $4$ and $2$ .

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Step 9.2.3.1

Factor $2$ out of $4 x^{2}$ .

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

Step 9.2.3.2

Cancel the common factors.

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Step 9.2.3.2.1

Factor $2$ out of $2$ .

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

Step 9.2.3.2.2

Cancel the common factor.

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

Step 9.2.3.2.3

Rewrite the expression.

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

Step 9.2.3.2.4

Divide $2 x^{2}$ by $1$ .

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

Step 10

Reorder terms.

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

Step 11

The answer is the antiderivative of the function $f (x) = x^{3} - 4 x^{2} + 4 x$ .

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