Calculus Examples

$\int (- 4 x^{6} + 6 x^{4} + 3 x^{3} + 6) d x$

Step 1

Remove parentheses.

$\int - 4 x^{6} + 6 x^{4} + 3 x^{3} + 6 d x$

Step 2

Split the single integral into multiple integrals.

$\int - 4 x^{6} d x + \int 6 x^{4} d x + \int 3 x^{3} d x + \int 6 d x$

Step 3

Since $- 4$ is constant with respect to $x$ , move $- 4$ out of the integral.

$- 4 \int x^{6} d x + \int 6 x^{4} d x + \int 3 x^{3} d x + \int 6 d x$

Step 4

By the Power Rule, the integral of $x^{6}$ with respect to $x$ is $\frac{1}{7} x^{7}$ .

$- 4 (\frac{1}{7} x^{7} + C) + \int 6 x^{4} d x + \int 3 x^{3} d x + \int 6 d x$

Step 5

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

$- 4 (\frac{1}{7} x^{7} + C) + 6 \int x^{4} d x + \int 3 x^{3} d x + \int 6 d x$

Step 6

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

$- 4 (\frac{1}{7} x^{7} + C) + 6 (\frac{1}{5} x^{5} + C) + \int 3 x^{3} d x + \int 6 d x$

Step 7

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

$- 4 (\frac{1}{7} x^{7} + C) + 6 (\frac{1}{5} x^{5} + C) + 3 \int x^{3} d x + \int 6 d x$

Step 8

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

$- 4 (\frac{1}{7} x^{7} + C) + 6 (\frac{1}{5} x^{5} + C) + 3 (\frac{1}{4} x^{4} + C) + \int 6 d x$

Step 9

Apply the constant rule.

$- 4 (\frac{1}{7} x^{7} + C) + 6 (\frac{1}{5} x^{5} + C) + 3 (\frac{1}{4} x^{4} + C) + 6 x + C$

Step 10

Simplify.

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Step 10.1

Simplify.

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Step 10.1.1

Combine $\frac{1}{7}$ and $x^{7}$ .

$- 4 (\frac{x^{7}}{7} + C) + 6 (\frac{1}{5} x^{5} + C) + 3 (\frac{1}{4} x^{4} + C) + 6 x + C$

Step 10.1.2

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

$- 4 (\frac{x^{7}}{7} + C) + 6 (\frac{x^{5}}{5} + C) + 3 (\frac{1}{4} x^{4} + C) + 6 x + C$

Step 10.1.3

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

$- 4 (\frac{x^{7}}{7} + C) + 6 (\frac{x^{5}}{5} + C) + 3 (\frac{x^{4}}{4} + C) + 6 x + C$

Step 10.2

Simplify.

$- \frac{4 x^{7}}{7} + \frac{6 x^{5}}{5} + \frac{3 x^{4}}{4} + 6 x + C$

Step 11

Reorder terms.

$- \frac{4}{7} x^{7} + \frac{6}{5} x^{5} + \frac{3}{4} x^{4} + 6 x + C$