Calculus Examples

$\frac{x^{3} - 1}{x^{2}}$

Step 1

Apply basic rules of exponents.

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Step 1.1

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

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

Step 1.2

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

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Step 1.2.1

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

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

Step 1.2.2

Multiply $2$ by $- 1$ .

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

Step 2

Multiply $(x^{3} - 1) x^{- 2}$ .

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

Step 3

Simplify.

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Step 3.1

Multiply $x^{3}$ by $x^{- 2}$ by adding the exponents.

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Step 3.1.1

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

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

Step 3.1.2

Subtract $2$ from $3$ .

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

Step 3.2

Simplify $x^{1}$ .

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

Step 3.3

Rewrite $- 1 x^{- 2}$ as $- x^{- 2}$ .

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

Step 4

Split the single integral into multiple integrals.

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

Simplify.

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Step 8.1

Simplify.

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

Step 8.2

Simplify.

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Step 8.2.1

Multiply $- 1$ by $- 1$ .

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

Step 8.2.2

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

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