Calculus Examples

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

Step 1

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

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

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.

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

Step 4

Split the single integral into multiple integrals.

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

Simplify the expression.

Tap for more steps...

Step 9.1

Simplify.

Tap for more steps...

Step 9.1.1

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

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

Step 9.1.2

Multiply $3$ by $- 1$ .

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

Step 9.2

Apply basic rules of exponents.

Tap for more steps...

Step 9.2.1

Move $x^{4}$ out of the denominator by raising it to the $- 1$ power.

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

Step 9.2.2

Multiply the exponents in ${(x^{4})}^{- 1}$ .

Tap for more steps...

Step 9.2.2.1

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

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

Step 9.2.2.2

Multiply $4$ by $- 1$ .

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

Step 10

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

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

Step 11

Simplify.

Tap for more steps...

Step 11.1

Simplify.

Tap for more steps...

Step 11.1.1

Combine $x^{- 3}$ and $\frac{1}{3}$ .

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

Step 11.1.2

Move $x^{- 3}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 11.2

Simplify.

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

Step 11.3

Simplify.

Tap for more steps...

Step 11.3.1

Multiply $- 1$ by $- 3$ .

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

Step 11.3.2

Combine $3$ and $\frac{1}{3 x^{3}}$ .

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

Step 11.3.3

Cancel the common factor of $3$ .

Tap for more steps...

Step 11.3.3.1

Cancel the common factor.

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

Step 11.3.3.2

Rewrite the expression.

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

Step 12

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

$F (x) =$ $x^{2} + \frac{1}{x^{3}} + C$