Calculus Examples

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

Step 1

Remove parentheses.

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

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

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

Step 4

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

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

Step 5

Since $- \frac{5}{2}$ is constant with respect to $x$ , move $- \frac{5}{2}$ out of the integral.

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

Step 6

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

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

Step 7

Apply the constant rule.

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

Step 8

Simplify.

$\frac{x^{6}}{9} - \frac{5 x^{2}}{4} + \frac{1}{2} x + C$

Step 9

Reorder terms.

$\frac{1}{9} x^{6} - \frac{5}{4} x^{2} + \frac{1}{2} x + C$