Calculus Examples

$\int (- \frac{4}{x^{3}} - \frac{8}{x^{5}}) d x$

Step 1

Remove parentheses.

$\int - \frac{4}{x^{3}} - \frac{8}{x^{5}} d x$

Step 2

Split the single integral into multiple integrals.

$\int - \frac{4}{x^{3}} d x + \int - \frac{8}{x^{5}} d x$

Step 3

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$- \int \frac{4}{x^{3}} d x + \int - \frac{8}{x^{5}} d x$

Step 4

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

$- (4 \int \frac{1}{x^{3}} d x) + \int - \frac{8}{x^{5}} d x$

Step 5

Simplify the expression.

Tap for more steps...

Step 5.1

Multiply $4$ by $- 1$ .

$- 4 \int \frac{1}{x^{3}} d x + \int - \frac{8}{x^{5}} d x$

Step 5.2

Move $x^{3}$ out of the denominator by raising it to the $- 1$ power.

$- 4 \int {(x^{3})}^{- 1} d x + \int - \frac{8}{x^{5}} d x$

Step 5.3

Multiply the exponents in ${(x^{3})}^{- 1}$ .

Tap for more steps...

Step 5.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$- 4 \int x^{3 \cdot - 1} d x + \int - \frac{8}{x^{5}} d x$

Step 5.3.2

Multiply $3$ by $- 1$ .

$- 4 \int x^{- 3} d x + \int - \frac{8}{x^{5}} d x$

Step 6

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

$- 4 (- \frac{1}{2} x^{- 2} + C) + \int - \frac{8}{x^{5}} d x$

Step 7

Simplify.

Tap for more steps...

Step 7.1

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

$- 4 (- \frac{x^{- 2}}{2} + C) + \int - \frac{8}{x^{5}} d x$

Step 7.2

Move $x^{- 2}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- 4 (- \frac{1}{2 x^{2}} + C) + \int - \frac{8}{x^{5}} d x$

Step 8

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$- 4 (- \frac{1}{2 x^{2}} + C) - \int \frac{8}{x^{5}} d x$

Step 9

Since $8$ is constant with respect to $x$ , move $8$ out of the integral.

$- 4 (- \frac{1}{2 x^{2}} + C) - (8 \int \frac{1}{x^{5}} d x)$

Step 10

Simplify the expression.

Tap for more steps...

Step 10.1

Multiply $8$ by $- 1$ .

$- 4 (- \frac{1}{2 x^{2}} + C) - 8 \int \frac{1}{x^{5}} d x$

Step 10.2

Move $x^{5}$ out of the denominator by raising it to the $- 1$ power.

$- 4 (- \frac{1}{2 x^{2}} + C) - 8 \int {(x^{5})}^{- 1} d x$

Step 10.3

Multiply the exponents in ${(x^{5})}^{- 1}$ .

Tap for more steps...

Step 10.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$- 4 (- \frac{1}{2 x^{2}} + C) - 8 \int x^{5 \cdot - 1} d x$

Step 10.3.2

Multiply $5$ by $- 1$ .

$- 4 (- \frac{1}{2 x^{2}} + C) - 8 \int x^{- 5} d x$

Step 11

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

$- 4 (- \frac{1}{2 x^{2}} + C) - 8 (- \frac{1}{4} x^{- 4} + C)$

Step 12

Simplify.

Tap for more steps...

Step 12.1

Simplify.

Tap for more steps...

Step 12.1.1

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

$- 4 (- \frac{1}{2 x^{2}} + C) - 8 (- \frac{x^{- 4}}{4} + C)$

Step 12.1.2

Move $x^{- 4}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- 4 (- \frac{1}{2 x^{2}} + C) - 8 (- \frac{1}{4 x^{4}} + C)$

Step 12.2

Simplify.

$\frac{2}{x^{2}} - 8 (- \frac{1}{4 x^{4}}) + C$

Step 12.3

Simplify.

Tap for more steps...

Step 12.3.1

Multiply $- 1$ by $- 8$ .

$\frac{2}{x^{2}} + 8 \frac{1}{4 x^{4}} + C$

Step 12.3.2

Combine $8$ and $\frac{1}{4 x^{4}}$ .

$\frac{2}{x^{2}} + \frac{8}{4 x^{4}} + C$

Step 12.3.3

Cancel the common factor of $8$ and $4$ .

Tap for more steps...

Step 12.3.3.1

Factor $4$ out of $8$ .

$\frac{2}{x^{2}} + \frac{4 \cdot 2}{4 x^{4}} + C$

Step 12.3.3.2

Cancel the common factors.

Tap for more steps...

Step 12.3.3.2.1

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

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

Step 12.3.3.2.2

Cancel the common factor.

$\frac{2}{x^{2}} + \frac{4 \cdot 2}{4 x^{4}} + C$

Step 12.3.3.2.3

Rewrite the expression.

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