Calculus Examples

$y = x^{2} - 1$ , $y = \frac{3}{x^{2} + 1}$

Step 1

Solve by substitution to find the intersection between the curves.

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Step 1.1

Eliminate the equal sides of each equation and combine.

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

Step 1.2

Solve $x^{2} - 1 = \frac{3}{x^{2} + 1}$ for $x$ .

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Step 1.2.1

Add $1$ to both sides of the equation.

$x^{2} = \frac{3}{x^{2} + 1} + 1$

Step 1.2.2

Find the LCD of the terms in the equation.

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Step 1.2.2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$1, x^{2} + 1, 1 + y = \frac{3}{x^{2} + 1}$

Step 1.2.2.2

Remove parentheses.

$1, x^{2} + 1, 1 + y = \frac{3}{x^{2} + 1}$

Step 1.2.2.3

The LCM of one and any expression is the expression.

$x^{2} + 1 + y = \frac{3}{x^{2} + 1}$

Step 1.2.3

Multiply each term in $x^{2} = \frac{3}{x^{2} + 1} + 1$ by $x^{2} + 1$ to eliminate the fractions.

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Step 1.2.3.1

Multiply each term in $x^{2} = \frac{3}{x^{2} + 1} + 1$ by $x^{2} + 1$ .

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

Step 1.2.3.2

Simplify the left side.

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Step 1.2.3.2.1

Apply the distributive property.

$x^{2} x^{2} + x^{2} \cdot 1 = \frac{3}{x^{2} + 1} (x^{2} + 1) + 1 (x^{2} + 1)$

Step 1.2.3.2.2

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

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Step 1.2.3.2.2.1

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

$x^{2 + 2} + x^{2} \cdot 1 = \frac{3}{x^{2} + 1} (x^{2} + 1) + 1 (x^{2} + 1)$

Step 1.2.3.2.2.2

Add $2$ and $2$ .

$x^{4} + x^{2} \cdot 1 = \frac{3}{x^{2} + 1} (x^{2} + 1) + 1 (x^{2} + 1)$

Step 1.2.3.2.3

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

$x^{4} + x^{2} = \frac{3}{x^{2} + 1} (x^{2} + 1) + 1 (x^{2} + 1)$

Step 1.2.3.3

Simplify the right side.

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Step 1.2.3.3.1

Simplify each term.

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Step 1.2.3.3.1.1

Cancel the common factor of $x^{2} + 1$ .

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Step 1.2.3.3.1.1.1

Cancel the common factor.

$x^{4} + x^{2} = \frac{3}{x^{2} + 1} (x^{2} + 1) + 1 (x^{2} + 1)$

Step 1.2.3.3.1.1.2

Rewrite the expression.

$x^{4} + x^{2} = 3 + 1 (x^{2} + 1)$

Step 1.2.3.3.1.2

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

$x^{4} + x^{2} = 3 + x^{2} + 1$

Step 1.2.3.3.2

Add $3$ and $1$ .

$x^{4} + x^{2} = x^{2} + 4$

Step 1.2.4

Solve the equation.

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Step 1.2.4.1

Move all terms containing $x$ to the left side of the equation.

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Step 1.2.4.1.1

Subtract $x^{2}$ from both sides of the equation.

$x^{4} + x^{2} - x^{2} = 4$

Step 1.2.4.1.2

Combine the opposite terms in $x^{4} + x^{2} - x^{2}$ .

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Step 1.2.4.1.2.1

Subtract $x^{2}$ from $x^{2}$ .

$x^{4} + 0 = 4$

Step 1.2.4.1.2.2

Add $x^{4}$ and $0$ .

$x^{4} = 4$

Step 1.2.4.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt[4]{4}$

Step 1.2.4.3

Simplify $\pm \sqrt[4]{4}$ .

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Step 1.2.4.3.1

Rewrite $4$ as $2^{2}$ .

$x = \pm \sqrt[4]{2^{2}}$

Step 1.2.4.3.2

Rewrite $\sqrt[4]{2^{2}}$ as $\sqrt{\sqrt{2^{2}}}$ .

$x = \pm \sqrt{\sqrt{2^{2}}}$

Step 1.2.4.3.3

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm \sqrt{2}$

Step 1.2.4.4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 1.2.4.4.1

First, use the positive value of the $\pm$ to find the first solution.

$x = \sqrt{2}$

Step 1.2.4.4.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \sqrt{2}$

Step 1.2.4.4.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = \sqrt{2}, - \sqrt{2}$

Step 1.3

Evaluate $y$ when $x = \sqrt{2}$ .

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Step 1.3.1

Substitute $\sqrt{2}$ for $x$ .

$y = \frac{3}{{(\sqrt{2})}^{2} + 1}$

Step 1.3.2

Substitute $\sqrt{2}$ for $x$ in $y = \frac{3}{{(\sqrt{2})}^{2} + 1}$ and solve for $y$ .

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Step 1.3.2.1

Remove parentheses.

$y = \frac{3}{{\sqrt{2}}^{2} + 1}$

Step 1.3.2.2

Simplify $\frac{3}{{\sqrt{2}}^{2} + 1}$ .

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Step 1.3.2.2.1

Simplify the denominator.

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Step 1.3.2.2.1.1

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 1.3.2.2.1.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$y = \frac{3}{{(2^{\frac{1}{2}})}^{2} + 1}$

Step 1.3.2.2.1.1.2

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

$y = \frac{3}{2^{\frac{1}{2} \cdot 2} + 1}$

Step 1.3.2.2.1.1.3

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

$y = \frac{3}{2^{\frac{2}{2}} + 1}$

Step 1.3.2.2.1.1.4

Cancel the common factor of $2$ .

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Step 1.3.2.2.1.1.4.1

Cancel the common factor.

$y = \frac{3}{2^{\frac{2}{2}} + 1}$

Step 1.3.2.2.1.1.4.2

Rewrite the expression.

$y = \frac{3}{2^{1} + 1}$

Step 1.3.2.2.1.1.5

Evaluate the exponent.

$y = \frac{3}{2 + 1}$

Step 1.3.2.2.1.2

Add $2$ and $1$ .

$y = \frac{3}{3}$

Step 1.3.2.2.2

Divide $3$ by $3$ .

$y = 1$

Step 1.4

The solution to the system is the complete set of ordered pairs that are valid solutions.

$(\sqrt{2}, 1)$

$(- \sqrt{2}, 1)$

$(\sqrt{2}, 1)$

$(- \sqrt{2}, 1)$

Step 2

The area of the region between the curves is defined as the integral of the upper curve minus the integral of the lower curve over each region. The regions are determined by the intersection points of the curves. This can be done algebraically or graphically.

$A r e a = \int_{- \sqrt{2}}^{\sqrt{2}} \frac{3}{x^{2} + 1} d x - \int_{- \sqrt{2}}^{\sqrt{2}} x^{2} - 1 d x$

Step 3

Integrate to find the area between $- \sqrt{2}$ and $\sqrt{2}$ .

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Step 3.1

Combine the integrals into a single integral.

$\int_{- \sqrt{2}}^{\sqrt{2}} \frac{3}{x^{2} + 1} - (x^{2} - 1) d x$

Step 3.2

Simplify each term.

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Step 3.2.1

Apply the distributive property.

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

Step 3.2.2

Multiply $- 1$ by $- 1$ .

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

$\int_{- \sqrt{2}}^{\sqrt{2}} \frac{3}{x^{2} + 1} - x^{2} + 1 d x$

Step 3.3

To write $- x^{2}$ as a fraction with a common denominator, multiply by $\frac{x^{2} + 1}{x^{2} + 1}$ .

$\int_{- \sqrt{2}}^{\sqrt{2}} \frac{3}{x^{2} + 1} - x^{2} \cdot \frac{x^{2} + 1}{x^{2} + 1} + 1 d x$

Step 3.4

Simplify terms.

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Step 3.4.1

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

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

Step 3.4.2

Combine the numerators over the common denominator.

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

$\int_{- \sqrt{2}}^{\sqrt{2}} \frac{3 - x^{2} (x^{2} + 1)}{x^{2} + 1} + 1 d x$

Step 3.5

Simplify the numerator.

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Step 3.5.1

Apply the distributive property.

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

Step 3.5.2

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

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Step 3.5.2.1

Move $x^{2}$ .

$\frac{3 - (x^{2} x^{2}) - x^{2} \cdot 1}{x^{2} + 1} + 1$

Step 3.5.2.2

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

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

Step 3.5.2.3

Add $2$ and $2$ .

$\frac{3 - x^{4} - x^{2} \cdot 1}{x^{2} + 1} + 1$

Step 3.5.3

Multiply $- 1$ by $1$ .

$\frac{3 - x^{4} - x^{2}}{x^{2} + 1} + 1$

Step 3.5.4

Reorder terms.

$\frac{- x^{4} - x^{2} + 3}{x^{2} + 1} + 1$

$\int_{- \sqrt{2}}^{\sqrt{2}} \frac{- x^{4} - x^{2} + 3}{x^{2} + 1} + 1 d x$

Step 3.6

Combine into one fraction.

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Step 3.6.1

Write $1$ as a fraction with a common denominator.

$\frac{- x^{4} - x^{2} + 3}{x^{2} + 1} + \frac{x^{2} + 1}{x^{2} + 1}$

Step 3.6.2

Combine the numerators over the common denominator.

$\frac{- x^{4} - x^{2} + 3 + x^{2} + 1}{x^{2} + 1}$

$\int_{- \sqrt{2}}^{\sqrt{2}} \frac{- x^{4} - x^{2} + 3 + x^{2} + 1}{x^{2} + 1} d x$

Step 3.7

Simplify the numerator.

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Step 3.7.1

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

$\frac{- x^{4} + 0 + 3 + 1}{x^{2} + 1}$

Step 3.7.2

Add $- x^{4}$ and $0$ .

$\frac{- x^{4} + 3 + 1}{x^{2} + 1}$

Step 3.7.3

Add $3$ and $1$ .

$\frac{- x^{4} + 4}{x^{2} + 1}$

Step 3.7.4

Rewrite $4$ as $2^{2}$ .

$\frac{- x^{4} + 2^{2}}{x^{2} + 1}$

Step 3.7.5

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

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

Step 3.7.6

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

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

Step 3.7.7

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 2$ and $b = x^{2}$ .

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

$\int_{- \sqrt{2}}^{\sqrt{2}} \frac{(2 + x^{2}) (2 - x^{2})}{x^{2} + 1} d x$

Step 3.8

Apply the distributive property.

$\int_{- \sqrt{2}}^{\sqrt{2}} \frac{2 (2 - x^{2}) + x^{2} (2 - x^{2})}{x^{2} + 1} d x$

Step 3.9

Apply the distributive property.

$\int_{- \sqrt{2}}^{\sqrt{2}} \frac{2 \cdot 2 + 2 (- x^{2}) + x^{2} (2 - x^{2})}{x^{2} + 1} d x$

Step 3.10

Apply the distributive property.

$\int_{- \sqrt{2}}^{\sqrt{2}} \frac{2 \cdot 2 + 2 (- x^{2}) + x^{2} \cdot 2 + x^{2} (- x^{2})}{x^{2} + 1} d x$

Step 3.11

Simplify the expression.

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Step 3.11.1

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

$\int_{- \sqrt{2}}^{\sqrt{2}} \frac{2 \cdot 2 + 2 \cdot - 1 x^{2} + 2 \cdot x^{2} + x^{2} (- x^{2})}{x^{2} + 1} d x$

Step 3.11.2

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

$\int_{- \sqrt{2}}^{\sqrt{2}} \frac{2 \cdot 2 + 2 \cdot - 1 x^{2} + 2 \cdot x^{2} - 1 \cdot x^{2} x^{2}}{x^{2} + 1} d x$

Step 3.11.3

Multiply $2$ by $2$ .

$\int_{- \sqrt{2}}^{\sqrt{2}} \frac{4 + 2 \cdot - 1 x^{2} + 2 x^{2} - 1 x^{2} x^{2}}{x^{2} + 1} d x$

Step 3.11.4

Multiply $2$ by $- 1$ .

$\int_{- \sqrt{2}}^{\sqrt{2}} \frac{4 - 2 x^{2} + 2 x^{2} - 1 x^{2} x^{2}}{x^{2} + 1} d x$

Step 3.12

Factor out negative.

$\int_{- \sqrt{2}}^{\sqrt{2}} \frac{4 - 2 x^{2} + 2 x^{2} - (x^{2} x^{2})}{x^{2} + 1} d x$

Step 3.13

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

$\int_{- \sqrt{2}}^{\sqrt{2}} \frac{4 - 2 x^{2} + 2 x^{2} - x^{2 + 2}}{x^{2} + 1} d x$

Step 3.14

Add $2$ and $2$ .

$\int_{- \sqrt{2}}^{\sqrt{2}} \frac{4 - 2 x^{2} + 2 x^{2} - x^{4}}{x^{2} + 1} d x$

Step 3.15

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

$\int_{- \sqrt{2}}^{\sqrt{2}} \frac{4 + 0 - x^{4}}{x^{2} + 1} d x$

Step 3.16

Simplify the expression.

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Step 3.16.1

Subtract $x^{4}$ from $0$ .

$\int_{- \sqrt{2}}^{\sqrt{2}} \frac{4 - x^{4}}{x^{2} + 1} d x$

Step 3.16.2

Reorder $4$ and $- x^{4}$ .

$\int_{- \sqrt{2}}^{\sqrt{2}} \frac{- x^{4} + 4}{x^{2} + 1} d x$

Step 3.17

Divide $- x^{4} + 4$ by $x^{2} + 1$ .

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Step 3.17.1

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$x^{2}$

$0 x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$4$

Step 3.17.2

Divide the highest order term in the dividend $- x^{4}$ by the highest order term in divisor $x^{2}$ .

$x^{2}$

$0 x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$4$

Step 3.17.3

Multiply the new quotient term by the divisor.

$x^{2}$

$0 x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$4$

$x^{4}$

$0$

$x^{2}$

Step 3.17.4

The expression needs to be subtracted from the dividend, so change all the signs in $- x^{4} + 0 - x^{2}$

$x^{2}$

$0 x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$4$

$x^{4}$

$0$

$x^{2}$

Step 3.17.5

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$x^{2}$

$0 x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$4$

$x^{4}$

$0$

$x^{2}$

Step 3.17.6

Pull the next term from the original dividend down into the current dividend.

$x^{2}$

$0 x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$4$

$x^{4}$

$0$

$x^{2}$

$0 x$

$4$

Step 3.17.7

Divide the highest order term in the dividend $x^{2}$ by the highest order term in divisor $x^{2}$ .

$x^{2}$

$0 x$

$1$

$x^{2}$

$0 x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$4$

$x^{4}$

$0$

$x^{2}$

$0 x$

$4$

Step 3.17.8

Multiply the new quotient term by the divisor.

$x^{2}$

$0 x$

$1$

$x^{2}$

$0 x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$4$

$x^{4}$

$0$

$x^{2}$

$0 x$

$4$

$x^{2}$

$0$

$1$

Step 3.17.9

The expression needs to be subtracted from the dividend, so change all the signs in $x^{2} + 0 + 1$

$x^{2}$

$0 x$

$1$

$x^{2}$

$0 x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$4$

$x^{4}$

$0$

$x^{2}$

$0 x$

$4$

$x^{2}$

$0$

$1$

Step 3.17.10

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$x^{2}$

$0 x$

$1$

$x^{2}$

$0 x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$4$

$x^{4}$

$0$

$x^{2}$

$0 x$

$4$

$x^{2}$

$0$

$1$

$3$

Step 3.17.11

The final answer is the quotient plus the remainder over the divisor.

$\int_{- \sqrt{2}}^{\sqrt{2}} - x^{2} + 1 + \frac{3}{x^{2} + 1} d x$

Step 3.18

Split the single integral into multiple integrals.

$\int_{- \sqrt{2}}^{\sqrt{2}} - x^{2} d x + \int_{- \sqrt{2}}^{\sqrt{2}} d x + \int_{- \sqrt{2}}^{\sqrt{2}} \frac{3}{x^{2} + 1} d x$

Step 3.19

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

$- \int_{- \sqrt{2}}^{\sqrt{2}} x^{2} d x + \int_{- \sqrt{2}}^{\sqrt{2}} d x + \int_{- \sqrt{2}}^{\sqrt{2}} \frac{3}{x^{2} + 1} d x$

Step 3.20

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

$- (\frac{1}{3} x^{3}]_{- \sqrt{2}}^{\sqrt{2}}) + \int_{- \sqrt{2}}^{\sqrt{2}} d x + \int_{- \sqrt{2}}^{\sqrt{2}} \frac{3}{x^{2} + 1} d x$

Step 3.21

Combine $\frac{1}{3}$ and $x^{3}$ .

$- (\frac{x^{3}}{3}]_{- \sqrt{2}}^{\sqrt{2}}) + \int_{- \sqrt{2}}^{\sqrt{2}} d x + \int_{- \sqrt{2}}^{\sqrt{2}} \frac{3}{x^{2} + 1} d x$

Step 3.22

Apply the constant rule.

$- (\frac{x^{3}}{3}]_{- \sqrt{2}}^{\sqrt{2}}) + x]_{- \sqrt{2}}^{\sqrt{2}} + \int_{- \sqrt{2}}^{\sqrt{2}} \frac{3}{x^{2} + 1} d x$

Step 3.23

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

$- (\frac{x^{3}}{3}]_{- \sqrt{2}}^{\sqrt{2}}) + x]_{- \sqrt{2}}^{\sqrt{2}} + 3 \int_{- \sqrt{2}}^{\sqrt{2}} \frac{1}{x^{2} + 1} d x$

Step 3.24

Simplify the expression.

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Step 3.24.1

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

$- (\frac{x^{3}}{3}]_{- \sqrt{2}}^{\sqrt{2}}) + x]_{- \sqrt{2}}^{\sqrt{2}} + 3 \int_{- \sqrt{2}}^{\sqrt{2}} \frac{1}{1 + x^{2}} d x$

Step 3.24.2

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

$- (\frac{x^{3}}{3}]_{- \sqrt{2}}^{\sqrt{2}}) + x]_{- \sqrt{2}}^{\sqrt{2}} + 3 \int_{- \sqrt{2}}^{\sqrt{2}} \frac{1}{1^{2} + x^{2}} d x$

Step 3.25

The integral of $\frac{1}{1^{2} + x^{2}}$ with respect to $x$ is $\arctan (x)]_{- \sqrt{2}}^{\sqrt{2}}$ .

$- (\frac{x^{3}}{3}]_{- \sqrt{2}}^{\sqrt{2}}) + x]_{- \sqrt{2}}^{\sqrt{2}} + 3 (\arctan (x)]_{- \sqrt{2}}^{\sqrt{2}})$

Step 3.26

Simplify the answer.

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Step 3.26.1

Substitute and simplify.

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Step 3.26.1.1

Evaluate $\frac{x^{3}}{3}$ at $\sqrt{2}$ and at $- \sqrt{2}$ .

$- ((\frac{{\sqrt{2}}^{3}}{3}) - \frac{{(- \sqrt{2})}^{3}}{3}) + x]_{- \sqrt{2}}^{\sqrt{2}} + 3 (\arctan (x)]_{- \sqrt{2}}^{\sqrt{2}})$

Step 3.26.1.2

Evaluate $x$ at $\sqrt{2}$ and at $- \sqrt{2}$ .

$- (\frac{{\sqrt{2}}^{3}}{3} - \frac{{(- \sqrt{2})}^{3}}{3}) + \sqrt{2} + \sqrt{2} + 3 (\arctan (x)]_{- \sqrt{2}}^{\sqrt{2}})$

Step 3.26.1.3

Evaluate $\arctan (x)$ at $\sqrt{2}$ and at $- \sqrt{2}$ .

$- (\frac{{\sqrt{2}}^{3}}{3} - \frac{{(- \sqrt{2})}^{3}}{3}) + \sqrt{2} + \sqrt{2} + 3 (\arctan (\sqrt{2}) - \arctan (- \sqrt{2}))$

Step 3.26.1.4

Simplify.

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Step 3.26.1.4.1

Rewrite ${\sqrt{2}}^{3}$ as $\sqrt{2^{3}}$ .

$- (\frac{\sqrt{2^{3}}}{3} - \frac{{(- \sqrt{2})}^{3}}{3}) + \sqrt{2} + \sqrt{2} + 3 (\arctan (\sqrt{2}) - \arctan (- \sqrt{2}))$

Step 3.26.1.4.2

Raise $2$ to the power of $3$ .

$- (\frac{\sqrt{8}}{3} - \frac{{(- \sqrt{2})}^{3}}{3}) + \sqrt{2} + \sqrt{2} + 3 (\arctan (\sqrt{2}) - \arctan (- \sqrt{2}))$

Step 3.26.1.4.3

Factor $- 1$ out of $- \sqrt{2}$ .

$- (\frac{\sqrt{8}}{3} - \frac{{(- (\sqrt{2}))}^{3}}{3}) + \sqrt{2} + \sqrt{2} + 3 (\arctan (\sqrt{2}) - \arctan (- \sqrt{2}))$

Step 3.26.1.4.4

Apply the product rule to $- (\sqrt{2})$ .

$- (\frac{\sqrt{8}}{3} - \frac{{(- 1)}^{3} {\sqrt{2}}^{3}}{3}) + \sqrt{2} + \sqrt{2} + 3 (\arctan (\sqrt{2}) - \arctan (- \sqrt{2}))$

Step 3.26.1.4.5

Raise $- 1$ to the power of $3$ .

$- (\frac{\sqrt{8}}{3} - \frac{- {\sqrt{2}}^{3}}{3}) + \sqrt{2} + \sqrt{2} + 3 (\arctan (\sqrt{2}) - \arctan (- \sqrt{2}))$

Step 3.26.1.4.6

Rewrite ${\sqrt{2}}^{3}$ as $\sqrt{2^{3}}$ .

$- (\frac{\sqrt{8}}{3} - \frac{- \sqrt{2^{3}}}{3}) + \sqrt{2} + \sqrt{2} + 3 (\arctan (\sqrt{2}) - \arctan (- \sqrt{2}))$

Step 3.26.1.4.7

Raise $2$ to the power of $3$ .

$- (\frac{\sqrt{8}}{3} - \frac{- \sqrt{8}}{3}) + \sqrt{2} + \sqrt{2} + 3 (\arctan (\sqrt{2}) - \arctan (- \sqrt{2}))$

Step 3.26.1.4.8

Move the negative in front of the fraction.

$- (\frac{\sqrt{8}}{3} - - \frac{\sqrt{8}}{3}) + \sqrt{2} + \sqrt{2} + 3 (\arctan (\sqrt{2}) - \arctan (- \sqrt{2}))$

Step 3.26.1.4.9

Multiply $- 1$ by $- 1$ .

$- (\frac{\sqrt{8}}{3} + 1 \frac{\sqrt{8}}{3}) + \sqrt{2} + \sqrt{2} + 3 (\arctan (\sqrt{2}) - \arctan (- \sqrt{2}))$

Step 3.26.1.4.10

Multiply $\frac{\sqrt{8}}{3}$ by $1$ .

$- (\frac{\sqrt{8}}{3} + \frac{\sqrt{8}}{3}) + \sqrt{2} + \sqrt{2} + 3 (\arctan (\sqrt{2}) - \arctan (- \sqrt{2}))$

Step 3.26.1.4.11

Combine the numerators over the common denominator.

$- \frac{\sqrt{8} + \sqrt{8}}{3} + \sqrt{2} + \sqrt{2} + 3 (\arctan (\sqrt{2}) - \arctan (- \sqrt{2}))$

Step 3.26.1.4.12

Add $\sqrt{8}$ and $\sqrt{8}$ .

$- \frac{2 \sqrt{8}}{3} + \sqrt{2} + \sqrt{2} + 3 (\arctan (\sqrt{2}) - \arctan (- \sqrt{2}))$

Step 3.26.1.4.13

Add $\sqrt{2}$ and $\sqrt{2}$ .

$- \frac{2 \sqrt{8}}{3} + 2 \sqrt{2} + 3 (\arctan (\sqrt{2}) - \arctan (- \sqrt{2}))$

Step 3.26.1.4.14

To write $3 (\arctan (\sqrt{2}) - \arctan (- \sqrt{2}))$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- \frac{2 \sqrt{8}}{3} + 3 (\arctan (\sqrt{2}) - \arctan (- \sqrt{2})) \cdot \frac{3}{3} + 2 \sqrt{2}$

Step 3.26.1.4.15

Combine $3 (\arctan (\sqrt{2}) - \arctan (- \sqrt{2}))$ and $\frac{3}{3}$ .

$- \frac{2 \sqrt{8}}{3} + \frac{3 (\arctan (\sqrt{2}) - \arctan (- \sqrt{2})) \cdot 3}{3} + 2 \sqrt{2}$

Step 3.26.1.4.16

Combine the numerators over the common denominator.

$\frac{- 2 \sqrt{8} + 3 (\arctan (\sqrt{2}) - \arctan (- \sqrt{2})) \cdot 3}{3} + 2 \sqrt{2}$

Step 3.26.1.4.17

Multiply $3$ by $3$ .

$\frac{- 2 \sqrt{8} + 9 (\arctan (\sqrt{2}) - \arctan (- \sqrt{2}))}{3} + 2 \sqrt{2}$

Step 3.26.2

Simplify.

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Step 3.26.2.1

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 3.26.2.1.1

Factor $4$ out of $8$ .

$\frac{- 2 \sqrt{4 (2)} + 9 (\arctan (\sqrt{2}) - \arctan (- \sqrt{2}))}{3} + 2 \sqrt{2}$

Step 3.26.2.1.2

Rewrite $4$ as $2^{2}$ .

$\frac{- 2 \sqrt{2^{2} \cdot 2} + 9 (\arctan (\sqrt{2}) - \arctan (- \sqrt{2}))}{3} + 2 \sqrt{2}$

Step 3.26.2.2

Pull terms out from under the radical.

$\frac{- 2 (2 \sqrt{2}) + 9 (\arctan (\sqrt{2}) - \arctan (- \sqrt{2}))}{3} + 2 \sqrt{2}$

Step 3.26.2.3

Multiply $2$ by $- 2$ .

$\frac{- 4 \sqrt{2} + 9 (\arctan (\sqrt{2}) - \arctan (- \sqrt{2}))}{3} + 2 \sqrt{2}$

Step 3.26.3

Simplify.

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Step 3.26.3.1

Evaluate $\arctan (- \sqrt{2})$ .

$\frac{- 4 \sqrt{2} + 9 (\arctan (\sqrt{2}) - - 0.95531661)}{3} + 2 \sqrt{2}$

Step 3.26.3.2

Multiply $- 1$ by $- 0.95531661$ .

$\frac{- 4 \sqrt{2} + 9 (\arctan (\sqrt{2}) + 0.95531661)}{3} + 2 \sqrt{2}$

Step 3.26.3.3

Evaluate $\arctan (\sqrt{2})$ .

$\frac{- 4 \sqrt{2} + 9 (0.95531661 + 0.95531661)}{3} + 2 \sqrt{2}$

Step 3.26.3.4

Add $0.95531661$ and $0.95531661$ .

$\frac{- 4 \sqrt{2} + 9 \cdot 1.91063323}{3} + 2 \sqrt{2}$

Step 3.26.3.5

Multiply $9$ by $1.91063323$ .

$\frac{- 4 \sqrt{2} + 17.19569912}{3} + 2 \sqrt{2}$

Step 3.26.3.6

Add $- 4 \sqrt{2}$ and $17.19569912$ .

$\frac{11.53884487}{3} + 2 \sqrt{2}$

Step 3.26.3.7

Divide $11.53884487$ by $3$ .

$3.84628162 + 2 \sqrt{2}$

Step 3.26.3.8

Add $3.84628162$ and $2 \sqrt{2}$ .

$6.67470875$

Step 4