Calculus Examples

$\int_{\frac{π}{4}}^{\frac{π}{2}} \sin^{3} (x) \cos (x) \sqrt{2 \sin^{2} (x) - 1} d x$

Step 1

Let $u_{1} = \sin (x)$ . Then $d u_{1} = \cos (x) d x$ , so $\frac{1}{\cos (x)} d u_{1} = d x$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

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Step 1.1

Let $u_{1} = \sin (x)$ . Find $\frac{d u_{1}}{d x}$ .

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Step 1.1.1

Differentiate $\sin (x)$ .

$\frac{d}{d x} [\sin (x)]$

Step 1.1.2

The derivative of $\sin (x)$ with respect to $x$ is $\cos (x)$ .

$\cos (x)$

Step 1.2

Substitute the lower limit in for $x$ in $u_{1} = \sin (x)$ .

$u_{lower} = \sin (\frac{π}{4})$

Step 1.3

The exact value of $\sin (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$u_{lower} = \frac{\sqrt{2}}{2}$

Step 1.4

Substitute the upper limit in for $x$ in $u_{1} = \sin (x)$ .

$u_{upper} = \sin (\frac{π}{2})$

Step 1.5

The exact value of $\sin (\frac{π}{2})$ is $1$ .

$u_{upper} = 1$

Step 1.6

The values found for $u_{lower}$ and $u_{upper}$ will be used to evaluate the definite integral.

$u_{lower} = \frac{\sqrt{2}}{2}$

$u_{upper} = 1$

Step 1.7

Rewrite the problem using $u_{1}$ , $d u_{1}$ , and the new limits of integration.

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

Step 2

Let $u_{1} = \frac{1}{\sqrt{2}} \sec (t)$ , where $- \frac{π}{2} \leq t \leq \frac{π}{2}$ . Then $d u_{1} = \frac{\sqrt{2} \sec (t) \tan (t)}{2} d t$ . Note that since $- \frac{π}{2} \leq t \leq \frac{π}{2}$ , $\frac{\sqrt{2} \sec (t) \tan (t)}{2}$ is positive.

$\int_{0}^{\frac{π}{4}} {(\frac{1}{\sqrt{2}} \sec (t))}^{3} \sqrt{{\sqrt{2}}^{2} {(\frac{1}{\sqrt{2}} \sec (t))}^{2} - 1} \frac{\sqrt{2} \sec (t) \tan (t)}{2} d t$

Step 3

Simplify terms.

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Step 3.1

Simplify $\sqrt{{\sqrt{2}}^{2} {(\frac{1}{\sqrt{2}} \sec (t))}^{2} - 1}$ .

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Step 3.1.1

Simplify each term.

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Step 3.1.1.1

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

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Step 3.1.1.1.1

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

$\int_{0}^{\frac{π}{4}} {(\frac{1}{\sqrt{2}} \sec (t))}^{3} \sqrt{{(2^{\frac{1}{2}})}^{2} {(\frac{1}{\sqrt{2}} \sec (t))}^{2} - 1} \frac{\sqrt{2} \sec (t) \tan (t)}{2} d t$

Step 3.1.1.1.2

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

$\int_{0}^{\frac{π}{4}} {(\frac{1}{\sqrt{2}} \sec (t))}^{3} \sqrt{2^{\frac{1}{2} \cdot 2} {(\frac{1}{\sqrt{2}} \sec (t))}^{2} - 1} \frac{\sqrt{2} \sec (t) \tan (t)}{2} d t$

Step 3.1.1.1.3

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

$\int_{0}^{\frac{π}{4}} {(\frac{1}{\sqrt{2}} \sec (t))}^{3} \sqrt{2^{\frac{2}{2}} {(\frac{1}{\sqrt{2}} \sec (t))}^{2} - 1} \frac{\sqrt{2} \sec (t) \tan (t)}{2} d t$

Step 3.1.1.1.4

Cancel the common factor of $2$ .

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Step 3.1.1.1.4.1

Cancel the common factor.

$\int_{0}^{\frac{π}{4}} {(\frac{1}{\sqrt{2}} \sec (t))}^{3} \sqrt{2^{\frac{2}{2}} {(\frac{1}{\sqrt{2}} \sec (t))}^{2} - 1} \frac{\sqrt{2} \sec (t) \tan (t)}{2} d t$

Step 3.1.1.1.4.2

Rewrite the expression.

$\int_{0}^{\frac{π}{4}} {(\frac{1}{\sqrt{2}} \sec (t))}^{3} \sqrt{2^{1} {(\frac{1}{\sqrt{2}} \sec (t))}^{2} - 1} \frac{\sqrt{2} \sec (t) \tan (t)}{2} d t$

Step 3.1.1.1.5

Evaluate the exponent.

$\int_{0}^{\frac{π}{4}} {(\frac{1}{\sqrt{2}} \sec (t))}^{3} \sqrt{2 {(\frac{1}{\sqrt{2}} \sec (t))}^{2} - 1} \frac{\sqrt{2} \sec (t) \tan (t)}{2} d t$

Step 3.1.1.2

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\int_{0}^{\frac{π}{4}} {(\frac{1}{\sqrt{2}} \sec (t))}^{3} \sqrt{2 {(\frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} \sec (t))}^{2} - 1} \frac{\sqrt{2} \sec (t) \tan (t)}{2} d t$

Step 3.1.1.3

Combine and simplify the denominator.

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Step 3.1.1.3.1

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\int_{0}^{\frac{π}{4}} {(\frac{1}{\sqrt{2}} \sec (t))}^{3} \sqrt{2 {(\frac{\sqrt{2}}{\sqrt{2} \sqrt{2}} \sec (t))}^{2} - 1} \frac{\sqrt{2} \sec (t) \tan (t)}{2} d t$

Step 3.1.1.3.2

Raise $\sqrt{2}$ to the power of $1$ .

$\int_{0}^{\frac{π}{4}} {(\frac{1}{\sqrt{2}} \sec (t))}^{3} \sqrt{2 {(\frac{\sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}} \sec (t))}^{2} - 1} \frac{\sqrt{2} \sec (t) \tan (t)}{2} d t$

Step 3.1.1.3.3

Raise $\sqrt{2}$ to the power of $1$ .

$\int_{0}^{\frac{π}{4}} {(\frac{1}{\sqrt{2}} \sec (t))}^{3} \sqrt{2 {(\frac{\sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}} \sec (t))}^{2} - 1} \frac{\sqrt{2} \sec (t) \tan (t)}{2} d t$

Step 3.1.1.3.4

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

$\int_{0}^{\frac{π}{4}} {(\frac{1}{\sqrt{2}} \sec (t))}^{3} \sqrt{2 {(\frac{\sqrt{2}}{{\sqrt{2}}^{1 + 1}} \sec (t))}^{2} - 1} \frac{\sqrt{2} \sec (t) \tan (t)}{2} d t$

Step 3.1.1.3.5

Add $1$ and $1$ .

$\int_{0}^{\frac{π}{4}} {(\frac{1}{\sqrt{2}} \sec (t))}^{3} \sqrt{2 {(\frac{\sqrt{2}}{{\sqrt{2}}^{2}} \sec (t))}^{2} - 1} \frac{\sqrt{2} \sec (t) \tan (t)}{2} d t$

Step 3.1.1.3.6

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

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Step 3.1.1.3.6.1

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

$\int_{0}^{\frac{π}{4}} {(\frac{1}{\sqrt{2}} \sec (t))}^{3} \sqrt{2 {(\frac{\sqrt{2}}{{(2^{\frac{1}{2}})}^{2}} \sec (t))}^{2} - 1} \frac{\sqrt{2} \sec (t) \tan (t)}{2} d t$

Step 3.1.1.3.6.2

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

$\int_{0}^{\frac{π}{4}} {(\frac{1}{\sqrt{2}} \sec (t))}^{3} \sqrt{2 {(\frac{\sqrt{2}}{2^{\frac{1}{2} \cdot 2}} \sec (t))}^{2} - 1} \frac{\sqrt{2} \sec (t) \tan (t)}{2} d t$

Step 3.1.1.3.6.3

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

$\int_{0}^{\frac{π}{4}} {(\frac{1}{\sqrt{2}} \sec (t))}^{3} \sqrt{2 {(\frac{\sqrt{2}}{2^{\frac{2}{2}}} \sec (t))}^{2} - 1} \frac{\sqrt{2} \sec (t) \tan (t)}{2} d t$

Step 3.1.1.3.6.4

Cancel the common factor of $2$ .

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Step 3.1.1.3.6.4.1

Cancel the common factor.

$\int_{0}^{\frac{π}{4}} {(\frac{1}{\sqrt{2}} \sec (t))}^{3} \sqrt{2 {(\frac{\sqrt{2}}{2^{\frac{2}{2}}} \sec (t))}^{2} - 1} \frac{\sqrt{2} \sec (t) \tan (t)}{2} d t$

Step 3.1.1.3.6.4.2

Rewrite the expression.

$\int_{0}^{\frac{π}{4}} {(\frac{1}{\sqrt{2}} \sec (t))}^{3} \sqrt{2 {(\frac{\sqrt{2}}{2^{1}} \sec (t))}^{2} - 1} \frac{\sqrt{2} \sec (t) \tan (t)}{2} d t$

Step 3.1.1.3.6.5

Evaluate the exponent.

$\int_{0}^{\frac{π}{4}} {(\frac{1}{\sqrt{2}} \sec (t))}^{3} \sqrt{2 {(\frac{\sqrt{2}}{2} \sec (t))}^{2} - 1} \frac{\sqrt{2} \sec (t) \tan (t)}{2} d t$

Step 3.1.1.4

Combine $\frac{\sqrt{2}}{2}$ and $\sec (t)$ .

$\int_{0}^{\frac{π}{4}} {(\frac{1}{\sqrt{2}} \sec (t))}^{3} \sqrt{2 {(\frac{\sqrt{2} \sec (t)}{2})}^{2} - 1} \frac{\sqrt{2} \sec (t) \tan (t)}{2} d t$

Step 3.1.1.5

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 3.1.1.5.1

Apply the product rule to $\frac{\sqrt{2} \sec (t)}{2}$ .

$\int_{0}^{\frac{π}{4}} {(\frac{1}{\sqrt{2}} \sec (t))}^{3} \sqrt{2 \frac{{(\sqrt{2} \sec (t))}^{2}}{2^{2}} - 1} \frac{\sqrt{2} \sec (t) \tan (t)}{2} d t$

Step 3.1.1.5.2

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

$\int_{0}^{\frac{π}{4}} {(\frac{1}{\sqrt{2}} \sec (t))}^{3} \sqrt{2 \frac{{\sqrt{2}}^{2} \sec^{2} (t)}{2^{2}} - 1} \frac{\sqrt{2} \sec (t) \tan (t)}{2} d t$

Step 3.1.1.6

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

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Step 3.1.1.6.1

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

$\int_{0}^{\frac{π}{4}} {(\frac{1}{\sqrt{2}} \sec (t))}^{3} \sqrt{2 \frac{{(2^{\frac{1}{2}})}^{2} \sec^{2} (t)}{2^{2}} - 1} \frac{\sqrt{2} \sec (t) \tan (t)}{2} d t$

Step 3.1.1.6.2

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

$\int_{0}^{\frac{π}{4}} {(\frac{1}{\sqrt{2}} \sec (t))}^{3} \sqrt{2 \frac{2^{\frac{1}{2} \cdot 2} \sec^{2} (t)}{2^{2}} - 1} \frac{\sqrt{2} \sec (t) \tan (t)}{2} d t$

Step 3.1.1.6.3

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

$\int_{0}^{\frac{π}{4}} {(\frac{1}{\sqrt{2}} \sec (t))}^{3} \sqrt{2 \frac{2^{\frac{2}{2}} \sec^{2} (t)}{2^{2}} - 1} \frac{\sqrt{2} \sec (t) \tan (t)}{2} d t$

Step 3.1.1.6.4

Cancel the common factor of $2$ .

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Step 3.1.1.6.4.1

Cancel the common factor.

$\int_{0}^{\frac{π}{4}} {(\frac{1}{\sqrt{2}} \sec (t))}^{3} \sqrt{2 \frac{2^{\frac{2}{2}} \sec^{2} (t)}{2^{2}} - 1} \frac{\sqrt{2} \sec (t) \tan (t)}{2} d t$

Step 3.1.1.6.4.2

Rewrite the expression.

$\int_{0}^{\frac{π}{4}} {(\frac{1}{\sqrt{2}} \sec (t))}^{3} \sqrt{2 \frac{2^{1} \sec^{2} (t)}{2^{2}} - 1} \frac{\sqrt{2} \sec (t) \tan (t)}{2} d t$

Step 3.1.1.6.5

Evaluate the exponent.

$\int_{0}^{\frac{π}{4}} {(\frac{1}{\sqrt{2}} \sec (t))}^{3} \sqrt{2 \frac{2 \sec^{2} (t)}{2^{2}} - 1} \frac{\sqrt{2} \sec (t) \tan (t)}{2} d t$

Step 3.1.1.7

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

$\int_{0}^{\frac{π}{4}} {(\frac{1}{\sqrt{2}} \sec (t))}^{3} \sqrt{2 \frac{2 \sec^{2} (t)}{4} - 1} \frac{\sqrt{2} \sec (t) \tan (t)}{2} d t$

Step 3.1.1.8

Cancel the common factor of $2$ .

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Step 3.1.1.8.1

Factor $2$ out of $4$ .

$\int_{0}^{\frac{π}{4}} {(\frac{1}{\sqrt{2}} \sec (t))}^{3} \sqrt{2 \frac{2 \sec^{2} (t)}{2 (2)} - 1} \frac{\sqrt{2} \sec (t) \tan (t)}{2} d t$

Step 3.1.1.8.2

Cancel the common factor.

$\int_{0}^{\frac{π}{4}} {(\frac{1}{\sqrt{2}} \sec (t))}^{3} \sqrt{2 \frac{2 \sec^{2} (t)}{2 \cdot 2} - 1} \frac{\sqrt{2} \sec (t) \tan (t)}{2} d t$

Step 3.1.1.8.3

Rewrite the expression.

$\int_{0}^{\frac{π}{4}} {(\frac{1}{\sqrt{2}} \sec (t))}^{3} \sqrt{\frac{2 \sec^{2} (t)}{2} - 1} \frac{\sqrt{2} \sec (t) \tan (t)}{2} d t$

Step 3.1.1.9

Cancel the common factor of $2$ .

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Step 3.1.1.9.1

Cancel the common factor.

$\int_{0}^{\frac{π}{4}} {(\frac{1}{\sqrt{2}} \sec (t))}^{3} \sqrt{\frac{2 \sec^{2} (t)}{2} - 1} \frac{\sqrt{2} \sec (t) \tan (t)}{2} d t$

Step 3.1.1.9.2

Divide $\sec^{2} (t)$ by $1$ .

$\int_{0}^{\frac{π}{4}} {(\frac{1}{\sqrt{2}} \sec (t))}^{3} \sqrt{\sec^{2} (t) - 1} \frac{\sqrt{2} \sec (t) \tan (t)}{2} d t$

Step 3.1.2

Apply pythagorean identity.

$\int_{0}^{\frac{π}{4}} {(\frac{1}{\sqrt{2}} \sec (t))}^{3} \sqrt{\tan^{2} (t)} \frac{\sqrt{2} \sec (t) \tan (t)}{2} d t$

Step 3.1.3

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

$\int_{0}^{\frac{π}{4}} {(\frac{1}{\sqrt{2}} \sec (t))}^{3} \tan (t) \frac{\sqrt{2} \sec (t) \tan (t)}{2} d t$

Step 3.2

Simplify the expression.

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Step 3.2.1

Simplify.

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Step 3.2.1.1

Combine $\frac{1}{\sqrt{2}}$ and $\sec (t)$ .

$\int_{0}^{\frac{π}{4}} {(\frac{\sec (t)}{\sqrt{2}})}^{3} \tan (t) \frac{\sqrt{2} \sec (t) \tan (t)}{2} d t$

Step 3.2.1.2

Combine $\frac{\sqrt{2} \sec (t) \tan (t)}{2}$ and $\tan (t)$ .

$\int_{0}^{\frac{π}{4}} {(\frac{\sec (t)}{\sqrt{2}})}^{3} \frac{\sqrt{2} \sec (t) \tan (t) \tan (t)}{2} d t$

Step 3.2.1.3

Raise $\tan (t)$ to the power of $1$ .

$\int_{0}^{\frac{π}{4}} {(\frac{\sec (t)}{\sqrt{2}})}^{3} \frac{\sqrt{2} \sec (t) (\tan^{1} (t) \tan (t))}{2} d t$

Step 3.2.1.4

Raise $\tan (t)$ to the power of $1$ .

$\int_{0}^{\frac{π}{4}} {(\frac{\sec (t)}{\sqrt{2}})}^{3} \frac{\sqrt{2} \sec (t) (\tan^{1} (t) \tan^{1} (t))}{2} d t$

Step 3.2.1.5

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

$\int_{0}^{\frac{π}{4}} {(\frac{\sec (t)}{\sqrt{2}})}^{3} \frac{\sqrt{2} \sec (t) {\tan (t)}^{1 + 1}}{2} d t$

Step 3.2.1.6

Add $1$ and $1$ .

$\int_{0}^{\frac{π}{4}} {(\frac{\sec (t)}{\sqrt{2}})}^{3} \frac{\sqrt{2} \sec (t) \tan^{2} (t)}{2} d t$

Step 3.2.2

Apply the product rule to $\frac{\sec (t)}{\sqrt{2}}$ .

$\int_{0}^{\frac{π}{4}} \frac{\sec^{3} (t)}{{\sqrt{2}}^{3}} \cdot \frac{\sqrt{2} \sec (t) \tan^{2} (t)}{2} d t$

Step 3.2.3

Simplify.

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Step 3.2.3.1

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

$\int_{0}^{\frac{π}{4}} \frac{\sec^{3} (t)}{\sqrt{2^{3}}} \cdot \frac{\sqrt{2} \sec (t) \tan^{2} (t)}{2} d t$

Step 3.2.3.2

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

$\int_{0}^{\frac{π}{4}} \frac{\sec^{3} (t)}{\sqrt{8}} \cdot \frac{\sqrt{2} \sec (t) \tan^{2} (t)}{2} d t$

Step 3.2.3.3

Multiply $\frac{\sec^{3} (t)}{\sqrt{8}}$ by $\frac{\sqrt{2} \sec (t) \tan^{2} (t)}{2}$ .

$\int_{0}^{\frac{π}{4}} \frac{\sec^{3} (t) (\sqrt{2} \sec (t) \tan^{2} (t))}{\sqrt{8} \cdot 2} d t$

Step 3.2.3.4

Multiply $\sec^{3} (t)$ by $\sec (t)$ by adding the exponents.

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Step 3.2.3.4.1

Move $\sec (t)$ .

$\int_{0}^{\frac{π}{4}} \frac{\sec (t) \sec^{3} (t) (\sqrt{2} \tan^{2} (t))}{\sqrt{8} \cdot 2} d t$

Step 3.2.3.4.2

Multiply $\sec (t)$ by $\sec^{3} (t)$ .

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Step 3.2.3.4.2.1

Raise $\sec (t)$ to the power of $1$ .

$\int_{0}^{\frac{π}{4}} \frac{\sec^{1} (t) \sec^{3} (t) (\sqrt{2} \tan^{2} (t))}{\sqrt{8} \cdot 2} d t$

Step 3.2.3.4.2.2

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

$\int_{0}^{\frac{π}{4}} \frac{{\sec (t)}^{1 + 3} (\sqrt{2} \tan^{2} (t))}{\sqrt{8} \cdot 2} d t$

Step 3.2.3.4.3

Add $1$ and $3$ .

$\int_{0}^{\frac{π}{4}} \frac{\sec^{4} (t) (\sqrt{2} \tan^{2} (t))}{\sqrt{8} \cdot 2} d t$

Step 3.2.3.5

Move $2$ to the left of $\sqrt{8}$ .

$\int_{0}^{\frac{π}{4}} \frac{\sec^{4} (t) \sqrt{2} \tan^{2} (t)}{2 \sqrt{8}} d t$

Step 4

Since $\frac{\sqrt{2}}{2 \sqrt{8}}$ is constant with respect to $t$ , move $\frac{\sqrt{2}}{2 \sqrt{8}}$ out of the integral.

$\frac{\sqrt{2}}{2 \sqrt{8}} \int_{0}^{\frac{π}{4}} \sec^{4} (t) \tan^{2} (t) d t$

Step 5

Simplify the expression.

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Step 5.1

Rewrite $4$ as $2$ plus $2$

$\frac{\sqrt{2}}{2 \sqrt{8}} \int_{0}^{\frac{π}{4}} {\sec (t)}^{2 + 2} \tan^{2} (t) d t$

Step 5.2

Rewrite ${\sec (t)}^{2 + 2}$ as $\sec^{2} (t) \sec^{2} (t)$ .

$\frac{\sqrt{2}}{2 \sqrt{8}} \int_{0}^{\frac{π}{4}} \sec^{2} (t) \sec^{2} (t) \tan^{2} (t) d t$

Step 6

Using the Pythagorean Identity, rewrite $\sec^{2} (t)$ as $1 + \tan^{2} (t)$ .

$\frac{\sqrt{2}}{2 \sqrt{8}} \int_{0}^{\frac{π}{4}} (1 + \tan^{2} (t)) \sec^{2} (t) \tan^{2} (t) d t$

Step 7

Let $u_{2} = \tan (t)$ . Then $d u_{2} = \sec^{2} (t) d t$ , so $\frac{1}{\sec^{2} (t)} d u_{2} = d t$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 7.1

Let $u_{2} = \tan (t)$ . Find $\frac{d u_{2}}{d t}$ .

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Step 7.1.1

Differentiate $\tan (t)$ .

$\frac{d}{d t} [\tan (t)]$

Step 7.1.2

The derivative of $\tan (t)$ with respect to $t$ is $\sec^{2} (t)$ .

$\sec^{2} (t)$

Step 7.2

Substitute the lower limit in for $t$ in $u_{2} = \tan (t)$ .

$u_{lower} = \tan (0)$

Step 7.3

The exact value of $\tan (0)$ is $0$ .

$u_{lower} = 0$

Step 7.4

Substitute the upper limit in for $t$ in $u_{2} = \tan (t)$ .

$u_{upper} = \tan (\frac{π}{4})$

Step 7.5

The exact value of $\tan (\frac{π}{4})$ is $1$ .

$u_{upper} = 1$

Step 7.6

The values found for $u_{lower}$ and $u_{upper}$ will be used to evaluate the definite integral.

$u_{lower} = 0$

$u_{upper} = 1$

Step 7.7

Rewrite the problem using $u_{2}$ , $d u_{2}$ , and the new limits of integration.

$\frac{\sqrt{2}}{2 \sqrt{8}} \int_{0}^{1} (1 + {u_{2}}^{2}) {u_{2}}^{2} d u_{2}$

Step 8

Multiply $(1 + {u_{2}}^{2}) {u_{2}}^{2}$ .

$\frac{\sqrt{2}}{2 \sqrt{8}} \int_{0}^{1} 1 {u_{2}}^{2} + {u_{2}}^{2} {u_{2}}^{2} d u_{2}$

Step 9

Simplify.

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Step 9.1

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

$\frac{\sqrt{2}}{2 \sqrt{8}} \int_{0}^{1} {u_{2}}^{2} + {u_{2}}^{2} {u_{2}}^{2} d u_{2}$

Step 9.2

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

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Step 9.2.1

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

$\frac{\sqrt{2}}{2 \sqrt{8}} \int_{0}^{1} {u_{2}}^{2} + {u_{2}}^{2 + 2} d u_{2}$

Step 9.2.2

Add $2$ and $2$ .

$\frac{\sqrt{2}}{2 \sqrt{8}} \int_{0}^{1} {u_{2}}^{2} + {u_{2}}^{4} d u_{2}$

Step 10

Split the single integral into multiple integrals.

$\frac{\sqrt{2}}{2 \sqrt{8}} (\int_{0}^{1} {u_{2}}^{2} d u_{2} + \int_{0}^{1} {u_{2}}^{4} d u_{2})$

Step 11

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

$\frac{\sqrt{2}}{2 \sqrt{8}} (\frac{1}{3} {u_{2}}^{3}]_{0}^{1} + \int_{0}^{1} {u_{2}}^{4} d u_{2})$

Step 12

By the Power Rule, the integral of ${u_{2}}^{4}$ with respect to $u_{2}$ is $\frac{1}{5} {u_{2}}^{5}$ .

$\frac{\sqrt{2}}{2 \sqrt{8}} (\frac{1}{3} {u_{2}}^{3}]_{0}^{1} + \frac{1}{5} {u_{2}}^{5}]_{0}^{1})$

Step 13

Combine $\frac{1}{3} {u_{2}}^{3}]_{0}^{1}$ and $\frac{1}{5} {u_{2}}^{5}]_{0}^{1}$ .

$\frac{\sqrt{2}}{2 \sqrt{8}} \frac{1}{3} {u_{2}}^{3} + \frac{1}{5} {u_{2}}^{5}]_{0}^{1}$

Step 14

Substitute and simplify.

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Step 14.1

Evaluate $\frac{1}{3} {u_{2}}^{3} + \frac{1}{5} {u_{2}}^{5}$ at $1$ and at $0$ .

$\frac{\sqrt{2}}{2 \sqrt{8}} ((\frac{1}{3} \cdot 1^{3} + \frac{1}{5} \cdot 1^{5}) - (\frac{1}{3} \cdot 0^{3} + \frac{1}{5} \cdot 0^{5}))$

Step 14.2

Simplify.

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Step 14.2.1

One to any power is one.

$\frac{\sqrt{2}}{2 \sqrt{8}} (\frac{1}{3} \cdot 1 + \frac{1}{5} \cdot 1^{5} - (\frac{1}{3} \cdot 0^{3} + \frac{1}{5} \cdot 0^{5}))$

Step 14.2.2

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

$\frac{\sqrt{2}}{2 \sqrt{8}} (\frac{1}{3} + \frac{1}{5} \cdot 1^{5} - (\frac{1}{3} \cdot 0^{3} + \frac{1}{5} \cdot 0^{5}))$

Step 14.2.3

One to any power is one.

$\frac{\sqrt{2}}{2 \sqrt{8}} (\frac{1}{3} + \frac{1}{5} \cdot 1 - (\frac{1}{3} \cdot 0^{3} + \frac{1}{5} \cdot 0^{5}))$

Step 14.2.4

Multiply $\frac{1}{5}$ by $1$ .

$\frac{\sqrt{2}}{2 \sqrt{8}} (\frac{1}{3} + \frac{1}{5} - (\frac{1}{3} \cdot 0^{3} + \frac{1}{5} \cdot 0^{5}))$

Step 14.2.5

To write $\frac{1}{3}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{\sqrt{2}}{2 \sqrt{8}} (\frac{1}{3} \cdot \frac{5}{5} + \frac{1}{5} - (\frac{1}{3} \cdot 0^{3} + \frac{1}{5} \cdot 0^{5}))$

Step 14.2.6

To write $\frac{1}{5}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{\sqrt{2}}{2 \sqrt{8}} (\frac{1}{3} \cdot \frac{5}{5} + \frac{1}{5} \cdot \frac{3}{3} - (\frac{1}{3} \cdot 0^{3} + \frac{1}{5} \cdot 0^{5}))$

Step 14.2.7

Write each expression with a common denominator of $15$ , by multiplying each by an appropriate factor of $1$ .

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Step 14.2.7.1

Multiply $\frac{1}{3}$ by $\frac{5}{5}$ .

$\frac{\sqrt{2}}{2 \sqrt{8}} (\frac{5}{3 \cdot 5} + \frac{1}{5} \cdot \frac{3}{3} - (\frac{1}{3} \cdot 0^{3} + \frac{1}{5} \cdot 0^{5}))$

Step 14.2.7.2

Multiply $3$ by $5$ .

$\frac{\sqrt{2}}{2 \sqrt{8}} (\frac{5}{15} + \frac{1}{5} \cdot \frac{3}{3} - (\frac{1}{3} \cdot 0^{3} + \frac{1}{5} \cdot 0^{5}))$

Step 14.2.7.3

Multiply $\frac{1}{5}$ by $\frac{3}{3}$ .

$\frac{\sqrt{2}}{2 \sqrt{8}} (\frac{5}{15} + \frac{3}{5 \cdot 3} - (\frac{1}{3} \cdot 0^{3} + \frac{1}{5} \cdot 0^{5}))$

Step 14.2.7.4

Multiply $5$ by $3$ .

$\frac{\sqrt{2}}{2 \sqrt{8}} (\frac{5}{15} + \frac{3}{15} - (\frac{1}{3} \cdot 0^{3} + \frac{1}{5} \cdot 0^{5}))$

Step 14.2.8

Combine the numerators over the common denominator.

$\frac{\sqrt{2}}{2 \sqrt{8}} (\frac{5 + 3}{15} - (\frac{1}{3} \cdot 0^{3} + \frac{1}{5} \cdot 0^{5}))$

Step 14.2.9

Add $5$ and $3$ .

$\frac{\sqrt{2}}{2 \sqrt{8}} (\frac{8}{15} - (\frac{1}{3} \cdot 0^{3} + \frac{1}{5} \cdot 0^{5}))$

Step 14.2.10

Raising $0$ to any positive power yields $0$ .

$\frac{\sqrt{2}}{2 \sqrt{8}} (\frac{8}{15} - (\frac{1}{3} \cdot 0 + \frac{1}{5} \cdot 0^{5}))$

Step 14.2.11

Multiply $\frac{1}{3}$ by $0$ .

$\frac{\sqrt{2}}{2 \sqrt{8}} (\frac{8}{15} - (0 + \frac{1}{5} \cdot 0^{5}))$

Step 14.2.12

Raising $0$ to any positive power yields $0$ .

$\frac{\sqrt{2}}{2 \sqrt{8}} (\frac{8}{15} - (0 + \frac{1}{5} \cdot 0))$

Step 14.2.13

Multiply $\frac{1}{5}$ by $0$ .

$\frac{\sqrt{2}}{2 \sqrt{8}} (\frac{8}{15} - (0 + 0))$

Step 14.2.14

Add $0$ and $0$ .

$\frac{\sqrt{2}}{2 \sqrt{8}} (\frac{8}{15} - 0)$

Step 14.2.15

Multiply $- 1$ by $0$ .

$\frac{\sqrt{2}}{2 \sqrt{8}} (\frac{8}{15} + 0)$

Step 14.2.16

Add $\frac{8}{15}$ and $0$ .

$\frac{\sqrt{2}}{2 \sqrt{8}} \cdot \frac{8}{15}$

Step 14.2.17

Multiply $\frac{\sqrt{2}}{2 \sqrt{8}}$ by $\frac{8}{15}$ .

$\frac{\sqrt{2} \cdot 8}{2 \sqrt{8} \cdot 15}$

Step 14.2.18

Multiply $15$ by $2$ .

$\frac{\sqrt{2} \cdot 8}{30 \sqrt{8}}$

Step 14.2.19

Move $8$ to the left of $\sqrt{2}$ .

$\frac{8 \cdot \sqrt{2}}{30 \sqrt{8}}$

Step 14.2.20

Cancel the common factor of $8$ and $30$ .

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Step 14.2.20.1

Factor $2$ out of $8 \sqrt{2}$ .

$\frac{2 (4 \sqrt{2})}{30 \sqrt{8}}$

Step 14.2.20.2

Cancel the common factors.

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Step 14.2.20.2.1

Factor $2$ out of $30 \sqrt{8}$ .

$\frac{2 (4 \sqrt{2})}{2 (15 \sqrt{8})}$

Step 14.2.20.2.2

Cancel the common factor.

$\frac{2 (4 \sqrt{2})}{2 (15 \sqrt{8})}$

Step 14.2.20.2.3

Rewrite the expression.

$\frac{4 \sqrt{2}}{15 \sqrt{8}}$

Step 15

Simplify.

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Step 15.1

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

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Step 15.1.1

Factor $4$ out of $8$ .

$\frac{4 \sqrt{2}}{15 \sqrt{4 (2)}}$

Step 15.1.2

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

$\frac{4 \sqrt{2}}{15 \sqrt{2^{2} \cdot 2}}$

Step 15.2

Pull terms out from under the radical.

$\frac{4 \sqrt{2}}{15 (2 \sqrt{2})}$

Step 15.3

Multiply $2$ by $15$ .

$\frac{4 \sqrt{2}}{30 \sqrt{2}}$

Step 15.4

Cancel the common factor of $4$ and $30$ .

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Step 15.4.1

Factor $2$ out of $4 \sqrt{2}$ .

$\frac{2 (2 \sqrt{2})}{30 \sqrt{2}}$

Step 15.4.2

Cancel the common factors.

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Step 15.4.2.1

Factor $2$ out of $30 \sqrt{2}$ .

$\frac{2 (2 \sqrt{2})}{2 (15 \sqrt{2})}$

Step 15.4.2.2

Cancel the common factor.

$\frac{2 (2 \sqrt{2})}{2 (15 \sqrt{2})}$

Step 15.4.2.3

Rewrite the expression.

$\frac{2 \sqrt{2}}{15 \sqrt{2}}$

Step 15.5

Cancel the common factor of $\sqrt{2}$ .

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Step 15.5.1

Cancel the common factor.

$\frac{2 \sqrt{2}}{15 \sqrt{2}}$

Step 15.5.2

Rewrite the expression.

$\frac{2}{15}$

Step 16

The result can be shown in multiple forms.

Exact Form:

$\frac{2}{15}$

Decimal Form:

$0.1\overline{3}$