Calculus Examples

$\frac{1}{x} - \frac{3}{x^{3}}$

Step 1

Write $\frac{1}{x} - \frac{3}{x^{3}}$ as a function.

$f (x) = \frac{1}{x} - \frac{3}{x^{3}}$

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.

$F (x) = \int \frac{1}{x} - \frac{3}{x^{3}} d x$

Step 4

Split the single integral into multiple integrals.

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

Simplify the expression.

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Step 8.1

Multiply $3$ by $- 1$ .

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

Step 8.2

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

$\ln (| x |) + C - 3 \int {(x^{3})}^{- 1} d x$

Step 8.3

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

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Step 8.3.1

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

$\ln (| x |) + C - 3 \int x^{3 \cdot - 1} d x$

Step 8.3.2

Multiply $3$ by $- 1$ .

$\ln (| x |) + C - 3 \int x^{- 3} d x$

Step 9

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

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

Step 10

Simplify.

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Step 10.1

Simplify.

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Step 10.1.1

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

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

Step 10.1.2

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

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

Step 10.2

Simplify.

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

Step 11

The answer is the antiderivative of the function $f (x) = \frac{1}{x} - \frac{3}{x^{3}}$ .

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