Calculus Examples

$\frac{2}{x^{3}} - \frac{4}{x^{2}}$

Step 1

Write $\frac{2}{x^{3}} - \frac{4}{x^{2}}$ as a function.

$f (x) = \frac{2}{x^{3}} - \frac{4}{x^{2}}$

Step 2

The function $F (x)$ can be found by finding the indefinite integral of the derivative $f (x)$ .

$F (x) = \int f (x) d x$

Step 3

Set up the integral to solve.

$F (x) = \int \frac{2}{x^{3}} - \frac{4}{x^{2}} d x$

Step 4

Split the single integral into multiple integrals.

$\int \frac{2}{x^{3}} d x + \int - \frac{4}{x^{2}} d x$

Step 5

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

$2 \int \frac{1}{x^{3}} d x + \int - \frac{4}{x^{2}} d x$

Step 6

Apply basic rules of exponents.

Tap for more steps...

Step 6.1

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

$2 \int {(x^{3})}^{- 1} d x + \int - \frac{4}{x^{2}} d x$

Step 6.2

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

Tap for more steps...

Step 6.2.1

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

$2 \int x^{3 \cdot - 1} d x + \int - \frac{4}{x^{2}} d x$

Step 6.2.2

Multiply $3$ by $- 1$ .

$2 \int x^{- 3} d x + \int - \frac{4}{x^{2}} d x$

Step 7

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

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

Step 8

Simplify.

Tap for more steps...

Step 8.1

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

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

Step 8.2

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

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

Step 9

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

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

Step 10

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

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

Step 11

Simplify the expression.

Tap for more steps...

Step 11.1

Multiply $4$ by $- 1$ .

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

Step 11.2

Move $x^{2}$ out of the denominator by raising it to the $- 1$ power.

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

Step 11.3

Multiply the exponents in ${(x^{2})}^{- 1}$ .

Tap for more steps...

Step 11.3.1

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

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

Step 11.3.2

Multiply $2$ by $- 1$ .

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

Step 12

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

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

Step 13

Simplify.

Tap for more steps...

Step 13.1

Simplify.

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

Step 13.2

Simplify.

Tap for more steps...

Step 13.2.1

Move the negative in front of the fraction.

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

Step 13.2.2

Multiply $- 1$ by $- 4$ .

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

Step 14

The answer is the antiderivative of the function $f (x) = \frac{2}{x^{3}} - \frac{4}{x^{2}}$ .

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