Calculus Examples

$\int (3 x^{4} - 5 x^{2} + x) d x$

Step 1

Remove parentheses.

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

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

Step 4

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

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

Step 5

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

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

Step 6

By the Power Rule, the integral of $x^{2}$ with respect to $x$ is $\frac{1}{3} x^{3}$ .

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

Step 7

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

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

Step 8

Simplify.

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Step 8.1

Simplify.

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Step 8.1.1

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

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

Step 8.1.2

Combine $\frac{1}{3}$ and $x^{3}$ .

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

Step 8.2

Simplify.

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

Step 9

Reorder terms.

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