Calculus Examples

$5 x^{4} - 2 x^{5}$

Step 1

Write $5 x^{4} - 2 x^{5}$ as a function.

$f (x) = 5 x^{4} - 2 x^{5}$

Step 2

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 3

Set up the integral to solve.

$F (x) = \int 5 x^{4} - 2 x^{5} d x$

Step 4

Split the single integral into multiple integrals.

$\int 5 x^{4} d x + \int - 2 x^{5} d x$

Step 5

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

$5 \int x^{4} d x + \int - 2 x^{5} d x$

Step 6

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

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

Step 7

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

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

Step 8

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

$5 (\frac{1}{5} x^{5} + C) - 2 (\frac{1}{6} x^{6} + C)$

Step 9

Simplify.

Tap for more steps...

Step 9.1

Simplify.

$5 (\frac{1}{5}) x^{5} - 2 (\frac{1}{6}) x^{6} + C$

Step 9.2

Simplify.

Tap for more steps...

Step 9.2.1

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

$\frac{5}{5} x^{5} - 2 (\frac{1}{6}) x^{6} + C$

Step 9.2.2

Cancel the common factor of $5$ .

Tap for more steps...

Step 9.2.2.1

Cancel the common factor.

$\frac{5}{5} x^{5} - 2 (\frac{1}{6}) x^{6} + C$

Step 9.2.2.2

Rewrite the expression.

$1 x^{5} - 2 (\frac{1}{6}) x^{6} + C$

Step 9.2.3

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

$x^{5} - 2 (\frac{1}{6}) x^{6} + C$

Step 9.2.4

Combine $- 2$ and $\frac{1}{6}$ .

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

Step 9.2.5

Cancel the common factor of $- 2$ and $6$ .

Tap for more steps...

Step 9.2.5.1

Factor $2$ out of $- 2$ .

$x^{5} + \frac{2 (- 1)}{6} x^{6} + C$

Step 9.2.5.2

Cancel the common factors.

Tap for more steps...

Step 9.2.5.2.1

Factor $2$ out of $6$ .

$x^{5} + \frac{2 \cdot - 1}{2 \cdot 3} x^{6} + C$

Step 9.2.5.2.2

Cancel the common factor.

$x^{5} + \frac{2 \cdot - 1}{2 \cdot 3} x^{6} + C$

Step 9.2.5.2.3

Rewrite the expression.

$x^{5} + \frac{- 1}{3} x^{6} + C$

Step 9.2.6

Move the negative in front of the fraction.

$x^{5} - \frac{1}{3} x^{6} + C$

Step 10

The answer is the antiderivative of the function $f (x) = 5 x^{4} - 2 x^{5}$ .

$F (x) =$ $x^{5} - \frac{1}{3} x^{6} + C$