Calculus Examples

$f (x) = x^{4} + 2 x^{3} - 8 x + 1$

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.

$F (x) = \int x^{4} + 2 x^{3} - 8 x + 1 d x$

Step 3

Split the single integral into multiple integrals.

$\int x^{4} d x + \int 2 x^{3} d x + \int - 8 x d x + \int d x$

Step 4

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

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

Step 5

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

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

Step 6

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

$\frac{1}{5} x^{5} + C + 2 (\frac{1}{4} x^{4} + C) + \int - 8 x d x + \int d x$

Step 7

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

$\frac{1}{5} x^{5} + C + 2 (\frac{1}{4} x^{4} + C) - 8 \int x d x + \int d x$

Step 8

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

$\frac{1}{5} x^{5} + C + 2 (\frac{1}{4} x^{4} + C) - 8 (\frac{1}{2} x^{2} + C) + \int d x$

Step 9

Apply the constant rule.

$\frac{1}{5} x^{5} + C + 2 (\frac{1}{4} x^{4} + C) - 8 (\frac{1}{2} x^{2} + C) + x + C$

Step 10

Simplify.

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Step 10.1

Simplify.

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Step 10.1.1

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

$\frac{1}{5} x^{5} + C + 2 (\frac{x^{4}}{4} + C) - 8 (\frac{1}{2} x^{2} + C) + x + C$

Step 10.1.2

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

$\frac{1}{5} x^{5} + C + 2 (\frac{x^{4}}{4} + C) - 8 (\frac{x^{2}}{2} + C) + x + C$

Step 10.2

Simplify.

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

Step 10.3

Reorder terms.

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

Step 11

The answer is the antiderivative of the function $f (x) = x^{4} + 2 x^{3} - 8 x + 1$ .

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

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