Calculus Examples

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

Step 1

Remove parentheses.

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

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

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

Step 4

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

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

Step 5

Apply the constant rule.

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

Simplify.

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Step 9.1

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

$2 (\frac{x^{3}}{3} + C) + 2 x + C + 5 (\frac{x^{4}}{4} + C)$

Step 9.2

Simplify.

$\frac{2 x^{3}}{3} + 2 x + 5 \frac{x^{4}}{4} + C$

Step 9.3

Combine fractions.

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Step 9.3.1

Combine $5$ and $\frac{x^{4}}{4}$ .

$\frac{2 x^{3}}{3} + 2 x + \frac{5 x^{4}}{4} + C$

Step 9.3.2

Reorder terms.

$\frac{2}{3} x^{3} + 2 x + \frac{5}{4} x^{4} + C$