Calculus Examples

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

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.

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

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, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 1.2.2

Multiply $2$ by $- 1$ .

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

Step 2

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

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

Step 3

Simplify.

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Step 3.1

Multiply $x^{2}$ 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.

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

Step 3.1.2

Subtract $2$ from $2$ .

$\int x^{0} + 1 x^{- 2} d x$

Step 3.2

Simplify $x^{0}$ .

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

Step 3.3

Multiply $x^{- 2}$ by $1$ .

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

Step 4

Split the single integral into multiple integrals.

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

Step 5

Apply the constant rule.

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

Step 6

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

$x + C - x^{- 1} + C$

Step 7

Simplify.

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