Calculus Examples

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

Step 1

Remove parentheses.

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

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

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

Step 4

Since $- \frac{1}{2}$ is constant with respect to $x$ , move $- \frac{1}{2}$ out of the integral.

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

Apply the constant rule.

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

Step 9

Simplify.

$\frac{x^{5}}{5} - \frac{x^{4}}{8} + \frac{x^{2}}{8} - 2 x + C$

Step 10

Reorder terms.

$\frac{1}{5} x^{5} - \frac{1}{8} x^{4} + \frac{1}{8} x^{2} - 2 x + C$