Calculus Examples

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

Step 1

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 2

Set up the integral to solve.

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

Step 3

Split the single integral into multiple integrals.

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

Step 4

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

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

Step 5

Apply basic rules of exponents.

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Step 5.1

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

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

Step 5.2

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

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Step 5.2.1

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

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

Step 5.2.2

Multiply $3$ by $- 1$ .

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

Step 6

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

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

Step 7

Simplify.

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Step 7.1

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

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

Step 7.2

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

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

Step 8

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

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

Step 9

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

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

Step 10

Multiply $3$ by $- 1$ .

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

Step 11

The integral of $\frac{1}{x}$ with respect to $x$ is $\ln (| x |)$ .

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

Step 12

Simplify.

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Step 12.1

Simplify.

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

Step 12.2

Move the negative in front of the fraction.

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

Step 13

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

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