Calculus Examples

$f (x) = 5 x^{4} - 4 x^{5}$

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.

$F (x) = \int 5 x^{4} - 4 x^{5} d x$

Step 3

Split the single integral into multiple integrals.

$\int 5 x^{4} d x + \int - 4 x^{5} d x$

Step 4

Since $5$ is constant with respect to $x$ , move $5$ out of the integral.

$5 \int x^{4} d x + \int - 4 x^{5} d x$

Step 5

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

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

Step 6

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

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

Step 7

By the Power Rule, the integral of $x^{5}$ with respect to $x$ is $\frac{1}{6} x^{6}$ .

$5 (\frac{1}{5} x^{5} + C) - 4 (\frac{1}{6} x^{6} + C)$

Step 8

Simplify.

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Step 8.1

Simplify.

$5 (\frac{1}{5}) x^{5} - 4 (\frac{1}{6}) x^{6} + C$

Step 8.2

Simplify.

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Step 8.2.1

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

$\frac{5}{5} x^{5} - 4 (\frac{1}{6}) x^{6} + C$

Step 8.2.2

Cancel the common factor of $5$ .

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Step 8.2.2.1

Cancel the common factor.

$\frac{5}{5} x^{5} - 4 (\frac{1}{6}) x^{6} + C$

Step 8.2.2.2

Rewrite the expression.

$1 x^{5} - 4 (\frac{1}{6}) x^{6} + C$

Step 8.2.3

Multiply $x^{5}$ by $1$ .

$x^{5} - 4 (\frac{1}{6}) x^{6} + C$

Step 8.2.4

Combine $- 4$ and $\frac{1}{6}$ .

$x^{5} + \frac{- 4}{6} x^{6} + C$

Step 8.2.5

Cancel the common factor of $- 4$ and $6$ .

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Step 8.2.5.1

Factor $2$ out of $- 4$ .

$x^{5} + \frac{2 (- 2)}{6} x^{6} + C$

Step 8.2.5.2

Cancel the common factors.

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Step 8.2.5.2.1

Factor $2$ out of $6$ .

$x^{5} + \frac{2 \cdot - 2}{2 \cdot 3} x^{6} + C$

Step 8.2.5.2.2

Cancel the common factor.

$x^{5} + \frac{2 \cdot - 2}{2 \cdot 3} x^{6} + C$

Step 8.2.5.2.3

Rewrite the expression.

$x^{5} + \frac{- 2}{3} x^{6} + C$

Step 8.2.6

Move the negative in front of the fraction.

$x^{5} - \frac{2}{3} x^{6} + C$

Step 9

The answer is the antiderivative of the function $f (x) = 5 x^{4} - 4 x^{5}$ .

$F (x) =$ $x^{5} - \frac{2}{3} x^{6} + C$