Calculus Examples

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

Step 1

Write $\frac{{(x^{2} - 1)}^{2}}{x^{2}}$ as a function.

$f (x) = \frac{{(x^{2} - 1)}^{2}}{x^{2}}$

Step 2

The function $F (x)$ can be found by finding the indefinite integral of the derivative $f (x)$ .

$F (x) = \int f (x) d x$

Step 3

Set up the integral to solve.

$F (x) = \int \frac{{(x^{2} - 1)}^{2}}{x^{2}} d x$

Step 4

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

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

Step 5

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

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Step 5.1

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

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

Step 5.2

Multiply $2$ by $- 1$ .

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

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

Step 6

Expand ${(x^{2} - 1)}^{2} x^{- 2}$ .

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Step 6.1

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

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

Step 6.2

Apply the distributive property.

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

Step 6.3

Apply the distributive property.

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

Step 6.4

Apply the distributive property.

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

Step 6.5

Apply the distributive property.

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

Step 6.6

Apply the distributive property.

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

Step 6.7

Apply the distributive property.

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

Step 6.8

Reorder $x^{2}$ and $- 1$ .

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

Step 6.9

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

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

Step 6.10

Add $2$ and $2$ .

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

Step 6.11

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

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

Step 6.12

Subtract $2$ from $4$ .

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

Step 6.13

Factor out negative.

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

Step 6.14

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

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

Step 6.15

Subtract $2$ from $2$ .

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

Step 6.16

Anything raised to $0$ is $1$ .

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

Step 6.17

Multiply $- 1$ by $1$ .

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

Step 6.18

Factor out negative.

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

Step 6.19

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

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

Step 6.20

Subtract $2$ from $2$ .

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

Step 6.21

Anything raised to $0$ is $1$ .

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

Step 6.22

Multiply $- 1$ by $1$ .

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

Step 6.23

Multiply $- 1$ by $- 1$ .

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

Step 6.24

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

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

Step 6.25

Subtract $1$ from $- 1$ .

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

Step 6.26

Reorder $- 2$ and $x^{- 2}$ .

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

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

Step 7

Split the single integral into multiple integrals.

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

Step 8

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

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

Step 9

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

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

Step 10

Apply the constant rule.

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

Step 11

Simplify.

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Step 11.1

Simplify.

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

Step 11.2

Reorder terms.

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

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

Step 12

The answer is the antiderivative of the function $f (x) = \frac{{(x^{2} - 1)}^{2}}{x^{2}}$ .

$F (x) =$ $\frac{1}{3} x^{3} - \frac{1}{x} - 2 x + C$

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$