Calculus Examples

$\int \frac{x^{5} - 2 x}{x^{4}} d x$

Step 1

Simplify the expression.

Tap for more steps...

Step 1.1

Simplify.

Tap for more steps...

Step 1.1.1

Factor $x$ out of $x^{5} - 2 x$ .

Tap for more steps...

Step 1.1.1.1

Factor $x$ out of $x^{5}$ .

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

Step 1.1.1.2

Factor $x$ out of $- 2 x$ .

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

Step 1.1.1.3

Factor $x$ out of $x \cdot x^{4} + x \cdot - 2$ .

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

Step 1.1.2

Cancel the common factors.

Tap for more steps...

Step 1.1.2.1

Factor $x$ out of $x^{4}$ .

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

Step 1.1.2.2

Cancel the common factor.

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

Step 1.1.2.3

Rewrite the expression.

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

Step 1.2

Apply basic rules of exponents.

Tap for more steps...

Step 1.2.1

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

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

Step 1.2.2

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

Tap for more steps...

Step 1.2.2.1

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

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

Step 1.2.2.2

Multiply $3$ by $- 1$ .

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

Step 2

Multiply $(x^{4} - 2) x^{- 3}$ .

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

Step 3

Simplify.

Tap for more steps...

Step 3.1

Multiply $x^{4}$ by $x^{- 3}$ by adding the exponents.

Tap for more steps...

Step 3.1.1

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

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

Step 3.1.2

Subtract $3$ from $4$ .

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

Step 3.2

Simplify $x^{1}$ .

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

Step 4

Split the single integral into multiple integrals.

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

Simplify.

Tap for more steps...

Step 8.1

Simplify.

Tap for more steps...

Step 8.1.1

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

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

Step 8.1.2

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

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

Step 8.2

Simplify.

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

Step 8.3

Simplify.

Tap for more steps...

Step 8.3.1

Multiply $- 1$ by $- 2$ .

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

Step 8.3.2

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

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

Step 8.3.3

Cancel the common factor of $2$ .

Tap for more steps...

Step 8.3.3.1

Cancel the common factor.

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

Step 8.3.3.2

Rewrite the expression.

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