Calculus Examples

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

Step 1

Apply basic rules of exponents.

Tap for more steps...

Step 1.1

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

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

Step 1.2

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

Tap for more steps...

Step 1.2.1

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

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

Step 1.2.2

Multiply $3$ by $- 1$ .

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

Step 2

Multiply $(x + 1) x^{- 3}$ .

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

Step 3

Simplify.

Tap for more steps...

Step 3.1

Multiply $x$ by $x^{- 3}$ by adding the exponents.

Tap for more steps...

Step 3.1.1

Multiply $x$ by $x^{- 3}$ .

Tap for more steps...

Step 3.1.1.1

Raise $x$ to the power of $1$ .

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

Step 3.1.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 3.1.2

Subtract $3$ from $1$ .

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

Step 3.2

Multiply $x^{- 3}$ by $1$ .

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

Step 4

Split the single integral into multiple integrals.

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

Step 5

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

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

Step 6

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

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

Step 7

Simplify.

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