Calculus Examples

$f (x) = 20 x^{4} - 4 x^{3} + 15 x^{2} - 14 x - 9$

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.

$F (x) = \int 20 x^{4} - 4 x^{3} + 15 x^{2} - 14 x - 9 d x$

Step 3

Split the single integral into multiple integrals.

$\int 20 x^{4} d x + \int - 4 x^{3} d x + \int 15 x^{2} d x + \int - 14 x d x + \int - 9 d x$

Step 4

Since $20$ is constant with respect to $x$ , move $20$ out of the integral.

$20 \int x^{4} d x + \int - 4 x^{3} d x + \int 15 x^{2} d x + \int - 14 x d x + \int - 9 d x$

Step 5

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

$20 (\frac{1}{5} x^{5} + C) + \int - 4 x^{3} d x + \int 15 x^{2} d x + \int - 14 x d x + \int - 9 d x$

Step 6

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

$20 (\frac{1}{5} x^{5} + C) - 4 \int x^{3} d x + \int 15 x^{2} d x + \int - 14 x d x + \int - 9 d x$

Step 7

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

$20 (\frac{1}{5} x^{5} + C) - 4 (\frac{1}{4} x^{4} + C) + \int 15 x^{2} d x + \int - 14 x d x + \int - 9 d x$

Step 8

Since $15$ is constant with respect to $x$ , move $15$ out of the integral.

$20 (\frac{1}{5} x^{5} + C) - 4 (\frac{1}{4} x^{4} + C) + 15 \int x^{2} d x + \int - 14 x d x + \int - 9 d x$

Step 9

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

$20 (\frac{1}{5} x^{5} + C) - 4 (\frac{1}{4} x^{4} + C) + 15 (\frac{1}{3} x^{3} + C) + \int - 14 x d x + \int - 9 d x$

Step 10

Since $- 14$ is constant with respect to $x$ , move $- 14$ out of the integral.

$20 (\frac{1}{5} x^{5} + C) - 4 (\frac{1}{4} x^{4} + C) + 15 (\frac{1}{3} x^{3} + C) - 14 \int x d x + \int - 9 d x$

Step 11

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

$20 (\frac{1}{5} x^{5} + C) - 4 (\frac{1}{4} x^{4} + C) + 15 (\frac{1}{3} x^{3} + C) - 14 (\frac{1}{2} x^{2} + C) + \int - 9 d x$

Step 12

Apply the constant rule.

$20 (\frac{1}{5} x^{5} + C) - 4 (\frac{1}{4} x^{4} + C) + 15 (\frac{1}{3} x^{3} + C) - 14 (\frac{1}{2} x^{2} + C) - 9 x + C$

Step 13

Simplify.

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Step 13.1

Simplify.

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Step 13.1.1

Combine $\frac{1}{5}$ and $x^{5}$ .

$20 (\frac{x^{5}}{5} + C) - 4 (\frac{1}{4} x^{4} + C) + 15 (\frac{1}{3} x^{3} + C) - 14 (\frac{1}{2} x^{2} + C) - 9 x + C$

Step 13.1.2

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

$20 (\frac{x^{5}}{5} + C) - 4 (\frac{x^{4}}{4} + C) + 15 (\frac{1}{3} x^{3} + C) - 14 (\frac{1}{2} x^{2} + C) - 9 x + C$

Step 13.1.3

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

$20 (\frac{x^{5}}{5} + C) - 4 (\frac{x^{4}}{4} + C) + 15 (\frac{x^{3}}{3} + C) - 14 (\frac{1}{2} x^{2} + C) - 9 x + C$

Step 13.1.4

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

$20 (\frac{x^{5}}{5} + C) - 4 (\frac{x^{4}}{4} + C) + 15 (\frac{x^{3}}{3} + C) - 14 (\frac{x^{2}}{2} + C) - 9 x + C$

Step 13.2

Simplify.

$4 x^{5} - x^{4} + 5 x^{3} - 7 x^{2} - 9 x + C$

Step 14

The answer is the antiderivative of the function $f (x) = 20 x^{4} - 4 x^{3} + 15 x^{2} - 14 x - 9$ .

$F (x) =$ $4 x^{5} - x^{4} + 5 x^{3} - 7 x^{2} - 9 x + C$