Calculus Examples

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

Step 1

Remove parentheses.

$\int 2 x^{5} + x - 6 x^{3} d x$

Step 2

Split the single integral into multiple integrals.

$\int 2 x^{5} d x + \int x d x + \int - 6 x^{3} d x$

Step 3

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

$2 \int x^{5} d x + \int x d x + \int - 6 x^{3} d x$

Step 4

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

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

Step 5

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

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

Step 6

Combine $\frac{1}{6}$ and $x^{6}$ .

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

Step 7

Since $- 6$ is constant with respect to $x$ , move $- 6$ out of the integral.

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

Step 8

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

$2 (\frac{x^{6}}{6} + C) + \frac{1}{2} x^{2} + C - 6 (\frac{1}{4} x^{4} + C)$

Step 9

Simplify.

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Step 9.1

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

$2 (\frac{x^{6}}{6} + C) + \frac{1}{2} x^{2} + C - 6 (\frac{x^{4}}{4} + C)$

Step 9.2

Simplify.

$\frac{x^{6}}{3} + \frac{x^{2}}{2} - 6 \frac{x^{4}}{4} + C$

Step 9.3

Simplify.

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Step 9.3.1

Combine $- 6$ and $\frac{x^{4}}{4}$ .

$\frac{x^{6}}{3} + \frac{x^{2}}{2} + \frac{- 6 x^{4}}{4} + C$

Step 9.3.2

Cancel the common factor of $- 6$ and $4$ .

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Step 9.3.2.1

Factor $2$ out of $- 6 x^{4}$ .

$\frac{x^{6}}{3} + \frac{x^{2}}{2} + \frac{2 (- 3 x^{4})}{4} + C$

Step 9.3.2.2

Cancel the common factors.

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Step 9.3.2.2.1

Factor $2$ out of $4$ .

$\frac{x^{6}}{3} + \frac{x^{2}}{2} + \frac{2 (- 3 x^{4})}{2 (2)} + C$

Step 9.3.2.2.2

Cancel the common factor.

$\frac{x^{6}}{3} + \frac{x^{2}}{2} + \frac{2 (- 3 x^{4})}{2 \cdot 2} + C$

Step 9.3.2.2.3

Rewrite the expression.

$\frac{x^{6}}{3} + \frac{x^{2}}{2} + \frac{- 3 x^{4}}{2} + C$

Step 9.3.3

Move the negative in front of the fraction.

$\frac{x^{6}}{3} + \frac{x^{2}}{2} - \frac{3 x^{4}}{2} + C$

Step 10

Reorder terms.

$\frac{1}{3} x^{6} + \frac{1}{2} x^{2} - \frac{3}{2} x^{4} + C$