Calculus Examples

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

Step 1

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

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

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.

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

Step 4

Apply the product rule to $\frac{x^{2} + 1}{x}$ .

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

Step 5

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 6

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

Tap for more steps...

Step 6.1

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 6.2

Multiply $2$ by $- 1$ .

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

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

Step 7

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

Tap for more steps...

Step 7.1

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

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

Step 7.2

Apply the distributive property.

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

Step 7.3

Apply the distributive property.

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

Step 7.4

Apply the distributive property.

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

Step 7.5

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 7.6

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 7.7

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 7.8

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 7.9

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 7.10

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 7.11

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 7.12

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 7.13

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

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

Step 7.14

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 7.15

Subtract $2$ from $2$ .

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

Step 7.16

Anything raised to $0$ is $1$ .

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

Step 7.17

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

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

Step 7.18

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 7.19

Subtract $2$ from $2$ .

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

Step 7.20

Anything raised to $0$ is $1$ .

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

Step 7.21

Multiply $1$ by $1$ .

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

Step 7.22

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

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

Step 7.23

Add $1$ and $1$ .

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

Step 7.24

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

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

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

Step 8

Split the single integral into multiple integrals.

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

Step 9

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

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

Step 10

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 11

Apply the constant rule.

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

Step 12

Simplify.

Tap for more steps...

Step 12.1

Simplify.

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

Step 12.2

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 13

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

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

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