Calculus Examples

$\frac{x^{5}}{20} - \frac{x^{4}}{4} - \frac{2 x^{3}}{3}$

Step 1

Split the single integral into multiple integrals.

$\int \frac{x^{5}}{20} d x + \int - \frac{x^{4}}{4} d x + \int - \frac{2 x^{3}}{3} d x$

Step 2

Since $\frac{1}{20}$ is constant with respect to $x$ , move $\frac{1}{20}$ out of the integral.

$\frac{1}{20} \int x^{5} d x + \int - \frac{x^{4}}{4} d x + \int - \frac{2 x^{3}}{3} d x$

Step 3

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

$\frac{1}{20} (\frac{1}{6} x^{6} + C) + \int - \frac{x^{4}}{4} d x + \int - \frac{2 x^{3}}{3} d x$

Step 4

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$\frac{1}{20} (\frac{1}{6} x^{6} + C) - \int \frac{x^{4}}{4} d x + \int - \frac{2 x^{3}}{3} d x$

Step 5

Since $\frac{1}{4}$ is constant with respect to $x$ , move $\frac{1}{4}$ out of the integral.

$\frac{1}{20} (\frac{1}{6} x^{6} + C) - (\frac{1}{4} \int x^{4} d x) + \int - \frac{2 x^{3}}{3} d x$

Step 6

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

$\frac{1}{20} (\frac{1}{6} x^{6} + C) - \frac{1}{4} (\frac{1}{5} x^{5} + C) + \int - \frac{2 x^{3}}{3} d x$

Step 7

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$\frac{1}{20} (\frac{1}{6} x^{6} + C) - \frac{1}{4} (\frac{1}{5} x^{5} + C) - \int \frac{2 x^{3}}{3} d x$

Step 8

Since $\frac{2}{3}$ is constant with respect to $x$ , move $\frac{2}{3}$ out of the integral.

$\frac{1}{20} (\frac{1}{6} x^{6} + C) - \frac{1}{4} (\frac{1}{5} x^{5} + C) - (\frac{2}{3} \int x^{3} d x)$

Step 9

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

$\frac{1}{20} (\frac{1}{6} x^{6} + C) - \frac{1}{4} (\frac{1}{5} x^{5} + C) - \frac{2}{3} (\frac{1}{4} x^{4} + C)$

Step 10

Simplify.

Tap for more steps...

Step 10.1

Simplify.

$\frac{x^{6}}{120} - \frac{x^{5}}{20} - \frac{2}{3} (\frac{1}{4} x^{4}) + C$

Step 10.2

Simplify.

Tap for more steps...

Step 10.2.1

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

$\frac{x^{6}}{120} - \frac{x^{5}}{20} - \frac{2}{3} \cdot \frac{x^{4}}{4} + C$

Step 10.2.2

Multiply $\frac{x^{4}}{4}$ by $\frac{2}{3}$ .

$\frac{x^{6}}{120} - \frac{x^{5}}{20} - \frac{x^{4} \cdot 2}{4 \cdot 3} + C$

Step 10.2.3

Multiply $4$ by $3$ .

$\frac{x^{6}}{120} - \frac{x^{5}}{20} - \frac{x^{4} \cdot 2}{12} + C$

Step 10.2.4

Move $2$ to the left of $x^{4}$ .

$\frac{x^{6}}{120} - \frac{x^{5}}{20} - \frac{2 \cdot x^{4}}{12} + C$

Step 10.2.5

Cancel the common factor of $2$ and $12$ .

Tap for more steps...

Step 10.2.5.1

Factor $2$ out of $2 x^{4}$ .

$\frac{x^{6}}{120} - \frac{x^{5}}{20} - \frac{2 (x^{4})}{12} + C$

Step 10.2.5.2

Cancel the common factors.

Tap for more steps...

Step 10.2.5.2.1

Factor $2$ out of $12$ .

$\frac{x^{6}}{120} - \frac{x^{5}}{20} - \frac{2 x^{4}}{2 \cdot 6} + C$

Step 10.2.5.2.2

Cancel the common factor.

$\frac{x^{6}}{120} - \frac{x^{5}}{20} - \frac{2 x^{4}}{2 \cdot 6} + C$

Step 10.2.5.2.3

Rewrite the expression.

$\frac{x^{6}}{120} - \frac{x^{5}}{20} - \frac{x^{4}}{6} + C$

Step 11

Reorder terms.

$\frac{1}{120} x^{6} - \frac{1}{20} x^{5} - \frac{1}{6} x^{4} + C$