Calculus Examples

$f (x) = x^{4} + 2 x^{2} - 8 x$

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.

$F (x) = \int x^{4} + 2 x^{2} - 8 x d x$

Step 3

Split the single integral into multiple integrals.

$\int x^{4} d x + \int 2 x^{2} d x + \int - 8 x d x$

Step 4

By the Power Rule, the integral of $x^{4}$ with respect to $x$ is $\frac{1}{5} x^{5}$ .

$\frac{1}{5} x^{5} + C + \int 2 x^{2} d x + \int - 8 x d x$

Step 5

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$\frac{1}{5} x^{5} + C + 2 \int x^{2} d x + \int - 8 x d x$

Step 6

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

$\frac{1}{5} x^{5} + C + 2 (\frac{1}{3} x^{3} + C) + \int - 8 x d x$

Step 7

Since $- 8$ is constant with respect to $x$ , move $- 8$ out of the integral.

$\frac{1}{5} x^{5} + C + 2 (\frac{1}{3} x^{3} + C) - 8 \int x d x$

Step 8

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

$\frac{1}{5} x^{5} + C + 2 (\frac{1}{3} x^{3} + C) - 8 (\frac{1}{2} x^{2} + C)$

Step 9

Simplify.

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Step 9.1

Simplify.

$\frac{x^{5}}{5} + \frac{2 x^{3}}{3} - 8 (\frac{1}{2} x^{2}) + C$

Step 9.2

Simplify.

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Step 9.2.1

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

$\frac{x^{5}}{5} + \frac{2 x^{3}}{3} - 8 \frac{x^{2}}{2} + C$

Step 9.2.2

Combine $- 8$ and $\frac{x^{2}}{2}$ .

$\frac{x^{5}}{5} + \frac{2 x^{3}}{3} + \frac{- 8 x^{2}}{2} + C$

Step 9.2.3

Cancel the common factor of $- 8$ and $2$ .

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Step 9.2.3.1

Factor $2$ out of $- 8 x^{2}$ .

$\frac{x^{5}}{5} + \frac{2 x^{3}}{3} + \frac{2 (- 4 x^{2})}{2} + C$

Step 9.2.3.2

Cancel the common factors.

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Step 9.2.3.2.1

Factor $2$ out of $2$ .

$\frac{x^{5}}{5} + \frac{2 x^{3}}{3} + \frac{2 (- 4 x^{2})}{2 (1)} + C$

Step 9.2.3.2.2

Cancel the common factor.

$\frac{x^{5}}{5} + \frac{2 x^{3}}{3} + \frac{2 (- 4 x^{2})}{2 \cdot 1} + C$

Step 9.2.3.2.3

Rewrite the expression.

$\frac{x^{5}}{5} + \frac{2 x^{3}}{3} + \frac{- 4 x^{2}}{1} + C$

Step 9.2.3.2.4

Divide $- 4 x^{2}$ by $1$ .

$\frac{x^{5}}{5} + \frac{2 x^{3}}{3} - 4 x^{2} + C$

Step 9.3

Reorder terms.

$\frac{1}{5} x^{5} + \frac{2}{3} x^{3} - 4 x^{2} + C$

Step 10

The answer is the antiderivative of the function $f (x) = x^{4} + 2 x^{2} - 8 x$ .

$F (x) =$ $\frac{1}{5} x^{5} + \frac{2}{3} x^{3} - 4 x^{2} + C$

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