Calculus Examples

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

Step 1

Remove parentheses.

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

Simplify.

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Step 9.1

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

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

Step 9.2

Simplify.

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

Step 9.3

Simplify.

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Step 9.3.1

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

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

Step 9.3.2

Cancel the common factor of $3$ .

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Step 9.3.2.1

Cancel the common factor.

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

Step 9.3.2.2

Divide $x^{3}$ by $1$ .

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

Step 9.4

Reorder terms.

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