Calculus Examples

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

Step 1

Remove parentheses.

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

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

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

Step 4

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

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

Step 5

Apply basic rules of exponents.

Tap for more steps...

Step 5.1

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

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

Step 5.2

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

Tap for more steps...

Step 5.2.1

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

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

Step 5.2.2

Multiply $2$ by $- 1$ .

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

Step 6

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

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

Step 7

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

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

Step 8

Apply basic rules of exponents.

Tap for more steps...

Step 8.1

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

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

Step 8.2

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

Tap for more steps...

Step 8.2.1

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

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

Step 8.2.2

Multiply $3$ by $- 1$ .

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

Step 9

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

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

Step 10

Simplify.

Tap for more steps...

Step 10.1

Simplify.

Tap for more steps...

Step 10.1.1

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

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

Step 10.1.2

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

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

Step 10.2

Simplify.

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

Step 10.3

Simplify.

Tap for more steps...

Step 10.3.1

Multiply $- 1$ by $3$ .

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

Step 10.3.2

Combine $- 3$ and $\frac{1}{2 x^{2}}$ .

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

Step 10.3.3

Move the negative in front of the fraction.

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