Calculus Examples

$\int (14 x^{6} + x^{2} - 4 x) d x$

Step 1

Remove parentheses.

$\int 14 x^{6} + x^{2} - 4 x d x$

Step 2

Split the single integral into multiple integrals.

$\int 14 x^{6} d x + \int x^{2} d x + \int - 4 x d x$

Step 3

Since $14$ is constant with respect to $x$ , move $14$ out of the integral.

$14 \int x^{6} d x + \int x^{2} d x + \int - 4 x d x$

Step 4

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

$14 (\frac{1}{7} x^{7} + C) + \int x^{2} d x + \int - 4 x d x$

Step 5

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

$14 (\frac{1}{7} x^{7} + C) + \frac{1}{3} x^{3} + C + \int - 4 x d x$

Step 6

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

$14 (\frac{x^{7}}{7} + C) + \frac{1}{3} x^{3} + C + \int - 4 x d x$

Step 7

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

$14 (\frac{x^{7}}{7} + C) + \frac{1}{3} x^{3} + C - 4 \int x d x$

Step 8

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

$14 (\frac{x^{7}}{7} + C) + \frac{1}{3} x^{3} + C - 4 (\frac{1}{2} x^{2} + C)$

Step 9

Simplify.

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Step 9.1

Combine $\frac{1}{2}$ and $x^{2}$ .

$14 (\frac{x^{7}}{7} + C) + \frac{1}{3} x^{3} + C - 4 (\frac{x^{2}}{2} + C)$

Step 9.2

Simplify.

$2 x^{7} + \frac{x^{3}}{3} - 4 \frac{x^{2}}{2} + C$

Step 9.3

Simplify.

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Step 9.3.1

Combine $- 4$ and $\frac{x^{2}}{2}$ .

$2 x^{7} + \frac{x^{3}}{3} + \frac{- 4 x^{2}}{2} + C$

Step 9.3.2

Cancel the common factor of $- 4$ and $2$ .

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Step 9.3.2.1

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

$2 x^{7} + \frac{x^{3}}{3} + \frac{2 (- 2 x^{2})}{2} + C$

Step 9.3.2.2

Cancel the common factors.

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Step 9.3.2.2.1

Factor $2$ out of $2$ .

$2 x^{7} + \frac{x^{3}}{3} + \frac{2 (- 2 x^{2})}{2 (1)} + C$

Step 9.3.2.2.2

Cancel the common factor.

$2 x^{7} + \frac{x^{3}}{3} + \frac{2 (- 2 x^{2})}{2 \cdot 1} + C$

Step 9.3.2.2.3

Rewrite the expression.

$2 x^{7} + \frac{x^{3}}{3} + \frac{- 2 x^{2}}{1} + C$

Step 9.3.2.2.4

Divide $- 2 x^{2}$ by $1$ .

$2 x^{7} + \frac{x^{3}}{3} - 2 x^{2} + C$

Step 9.4

Reorder terms.

$2 x^{7} + \frac{1}{3} x^{3} - 2 x^{2} + C$