Calculus Examples

$\int \frac{3 x^{2}}{\sqrt{2 x^{2} + 5}} d x$

Step 1

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

$3 \int \frac{x^{2}}{\sqrt{2 x^{2} + 5}} d x$

Step 2

Let $x = \frac{\sqrt{5}}{\sqrt{2}} \tan (t)$ , where $- \frac{π}{2} \leq t \leq \frac{π}{2}$ . Then $d x = \frac{\sqrt{10} \sec^{2} (t)}{2} d t$ . Note that since $- \frac{π}{2} \leq t \leq \frac{π}{2}$ , $\frac{\sqrt{10} \sec^{2} (t)}{2}$ is positive.

$3 \int \frac{{(\frac{\sqrt{5}}{\sqrt{2}} \tan (t))}^{2}}{\sqrt{{\sqrt{2}}^{2} {(\frac{\sqrt{5}}{\sqrt{2}} \tan (t))}^{2} + {\sqrt{5}}^{2}}} \cdot \frac{\sqrt{10} \sec^{2} (t)}{2} d t$

Step 3

Simplify terms.

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Step 3.1

Simplify $\sqrt{{\sqrt{2}}^{2} {(\frac{\sqrt{5}}{\sqrt{2}} \tan (t))}^{2} + {\sqrt{5}}^{2}}$ .

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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 $2^{\frac{1}{2}}$ .

$3 \int \frac{{(\frac{\sqrt{5}}{\sqrt{2}} \tan (t))}^{2}}{\sqrt{{(2^{\frac{1}{2}})}^{2} {(\frac{\sqrt{5}}{\sqrt{2}} \tan (t))}^{2} + {\sqrt{5}}^{2}}} \cdot \frac{\sqrt{10} \sec^{2} (t)}{2} d t$

Step 3.1.1.1.2

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

$3 \int \frac{{(\frac{\sqrt{5}}{\sqrt{2}} \tan (t))}^{2}}{\sqrt{2^{\frac{1}{2} \cdot 2} {(\frac{\sqrt{5}}{\sqrt{2}} \tan (t))}^{2} + {\sqrt{5}}^{2}}} \cdot \frac{\sqrt{10} \sec^{2} (t)}{2} d t$

Step 3.1.1.1.3

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

$3 \int \frac{{(\frac{\sqrt{5}}{\sqrt{2}} \tan (t))}^{2}}{\sqrt{2^{\frac{2}{2}} {(\frac{\sqrt{5}}{\sqrt{2}} \tan (t))}^{2} + {\sqrt{5}}^{2}}} \cdot \frac{\sqrt{10} \sec^{2} (t)}{2} d t$

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.

$3 \int \frac{{(\frac{\sqrt{5}}{\sqrt{2}} \tan (t))}^{2}}{\sqrt{2^{\frac{2}{2}} {(\frac{\sqrt{5}}{\sqrt{2}} \tan (t))}^{2} + {\sqrt{5}}^{2}}} \cdot \frac{\sqrt{10} \sec^{2} (t)}{2} d t$

Step 3.1.1.1.4.2

Rewrite the expression.

$3 \int \frac{{(\frac{\sqrt{5}}{\sqrt{2}} \tan (t))}^{2}}{\sqrt{2^{1} {(\frac{\sqrt{5}}{\sqrt{2}} \tan (t))}^{2} + {\sqrt{5}}^{2}}} \cdot \frac{\sqrt{10} \sec^{2} (t)}{2} d t$

Step 3.1.1.1.5

Evaluate the exponent.

$3 \int \frac{{(\frac{\sqrt{5}}{\sqrt{2}} \tan (t))}^{2}}{\sqrt{2 {(\frac{\sqrt{5}}{\sqrt{2}} \tan (t))}^{2} + {\sqrt{5}}^{2}}} \cdot \frac{\sqrt{10} \sec^{2} (t)}{2} d t$

Step 3.1.1.2

Multiply $\frac{\sqrt{5}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$3 \int \frac{{(\frac{\sqrt{5}}{\sqrt{2}} \tan (t))}^{2}}{\sqrt{2 {(\frac{\sqrt{5}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} \tan (t))}^{2} + {\sqrt{5}}^{2}}} \cdot \frac{\sqrt{10} \sec^{2} (t)}{2} d t$

Step 3.1.1.3

Combine and simplify the denominator.

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Step 3.1.1.3.1

Multiply $\frac{\sqrt{5}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$3 \int \frac{{(\frac{\sqrt{5}}{\sqrt{2}} \tan (t))}^{2}}{\sqrt{2 {(\frac{\sqrt{5} \sqrt{2}}{\sqrt{2} \sqrt{2}} \tan (t))}^{2} + {\sqrt{5}}^{2}}} \cdot \frac{\sqrt{10} \sec^{2} (t)}{2} d t$

Step 3.1.1.3.2

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

$3 \int \frac{{(\frac{\sqrt{5}}{\sqrt{2}} \tan (t))}^{2}}{\sqrt{2 {(\frac{\sqrt{5} \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}} \tan (t))}^{2} + {\sqrt{5}}^{2}}} \cdot \frac{\sqrt{10} \sec^{2} (t)}{2} d t$

Step 3.1.1.3.3

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

$3 \int \frac{{(\frac{\sqrt{5}}{\sqrt{2}} \tan (t))}^{2}}{\sqrt{2 {(\frac{\sqrt{5} \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}} \tan (t))}^{2} + {\sqrt{5}}^{2}}} \cdot \frac{\sqrt{10} \sec^{2} (t)}{2} d t$

Step 3.1.1.3.4

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

$3 \int \frac{{(\frac{\sqrt{5}}{\sqrt{2}} \tan (t))}^{2}}{\sqrt{2 {(\frac{\sqrt{5} \sqrt{2}}{{\sqrt{2}}^{1 + 1}} \tan (t))}^{2} + {\sqrt{5}}^{2}}} \cdot \frac{\sqrt{10} \sec^{2} (t)}{2} d t$

Step 3.1.1.3.5

Add $1$ and $1$ .

$3 \int \frac{{(\frac{\sqrt{5}}{\sqrt{2}} \tan (t))}^{2}}{\sqrt{2 {(\frac{\sqrt{5} \sqrt{2}}{{\sqrt{2}}^{2}} \tan (t))}^{2} + {\sqrt{5}}^{2}}} \cdot \frac{\sqrt{10} \sec^{2} (t)}{2} d t$

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 $2^{\frac{1}{2}}$ .

$3 \int \frac{{(\frac{\sqrt{5}}{\sqrt{2}} \tan (t))}^{2}}{\sqrt{2 {(\frac{\sqrt{5} \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}} \tan (t))}^{2} + {\sqrt{5}}^{2}}} \cdot \frac{\sqrt{10} \sec^{2} (t)}{2} d t$

Step 3.1.1.3.6.2

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

$3 \int \frac{{(\frac{\sqrt{5}}{\sqrt{2}} \tan (t))}^{2}}{\sqrt{2 {(\frac{\sqrt{5} \sqrt{2}}{2^{\frac{1}{2} \cdot 2}} \tan (t))}^{2} + {\sqrt{5}}^{2}}} \cdot \frac{\sqrt{10} \sec^{2} (t)}{2} d t$

Step 3.1.1.3.6.3

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

$3 \int \frac{{(\frac{\sqrt{5}}{\sqrt{2}} \tan (t))}^{2}}{\sqrt{2 {(\frac{\sqrt{5} \sqrt{2}}{2^{\frac{2}{2}}} \tan (t))}^{2} + {\sqrt{5}}^{2}}} \cdot \frac{\sqrt{10} \sec^{2} (t)}{2} d t$

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.

$3 \int \frac{{(\frac{\sqrt{5}}{\sqrt{2}} \tan (t))}^{2}}{\sqrt{2 {(\frac{\sqrt{5} \sqrt{2}}{2^{\frac{2}{2}}} \tan (t))}^{2} + {\sqrt{5}}^{2}}} \cdot \frac{\sqrt{10} \sec^{2} (t)}{2} d t$

Step 3.1.1.3.6.4.2

Rewrite the expression.

$3 \int \frac{{(\frac{\sqrt{5}}{\sqrt{2}} \tan (t))}^{2}}{\sqrt{2 {(\frac{\sqrt{5} \sqrt{2}}{2^{1}} \tan (t))}^{2} + {\sqrt{5}}^{2}}} \cdot \frac{\sqrt{10} \sec^{2} (t)}{2} d t$

Step 3.1.1.3.6.5

Evaluate the exponent.

$3 \int \frac{{(\frac{\sqrt{5}}{\sqrt{2}} \tan (t))}^{2}}{\sqrt{2 {(\frac{\sqrt{5} \sqrt{2}}{2} \tan (t))}^{2} + {\sqrt{5}}^{2}}} \cdot \frac{\sqrt{10} \sec^{2} (t)}{2} d t$

Step 3.1.1.4

Simplify the numerator.

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Step 3.1.1.4.1

Combine using the product rule for radicals.

$3 \int \frac{{(\frac{\sqrt{5}}{\sqrt{2}} \tan (t))}^{2}}{\sqrt{2 {(\frac{\sqrt{5 \cdot 2}}{2} \tan (t))}^{2} + {\sqrt{5}}^{2}}} \cdot \frac{\sqrt{10} \sec^{2} (t)}{2} d t$

Step 3.1.1.4.2

Multiply $5$ by $2$ .

$3 \int \frac{{(\frac{\sqrt{5}}{\sqrt{2}} \tan (t))}^{2}}{\sqrt{2 {(\frac{\sqrt{10}}{2} \tan (t))}^{2} + {\sqrt{5}}^{2}}} \cdot \frac{\sqrt{10} \sec^{2} (t)}{2} d t$

Step 3.1.1.5

Combine $\frac{\sqrt{10}}{2}$ and $\tan (t)$ .

$3 \int \frac{{(\frac{\sqrt{5}}{\sqrt{2}} \tan (t))}^{2}}{\sqrt{2 {(\frac{\sqrt{10} \tan (t)}{2})}^{2} + {\sqrt{5}}^{2}}} \cdot \frac{\sqrt{10} \sec^{2} (t)}{2} d t$

Step 3.1.1.6

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

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Step 3.1.1.6.1

Apply the product rule to $\frac{\sqrt{10} \tan (t)}{2}$ .

$3 \int \frac{{(\frac{\sqrt{5}}{\sqrt{2}} \tan (t))}^{2}}{\sqrt{2 \frac{{(\sqrt{10} \tan (t))}^{2}}{2^{2}} + {\sqrt{5}}^{2}}} \cdot \frac{\sqrt{10} \sec^{2} (t)}{2} d t$

Step 3.1.1.6.2

Apply the product rule to $\sqrt{10} \tan (t)$ .

$3 \int \frac{{(\frac{\sqrt{5}}{\sqrt{2}} \tan (t))}^{2}}{\sqrt{2 \frac{{\sqrt{10}}^{2} \tan^{2} (t)}{2^{2}} + {\sqrt{5}}^{2}}} \cdot \frac{\sqrt{10} \sec^{2} (t)}{2} d t$

Step 3.1.1.7

Rewrite ${\sqrt{10}}^{2}$ as $10$ .

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Step 3.1.1.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{10}$ as $10^{\frac{1}{2}}$ .

$3 \int \frac{{(\frac{\sqrt{5}}{\sqrt{2}} \tan (t))}^{2}}{\sqrt{2 \frac{{(10^{\frac{1}{2}})}^{2} \tan^{2} (t)}{2^{2}} + {\sqrt{5}}^{2}}} \cdot \frac{\sqrt{10} \sec^{2} (t)}{2} d t$

Step 3.1.1.7.2

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

$3 \int \frac{{(\frac{\sqrt{5}}{\sqrt{2}} \tan (t))}^{2}}{\sqrt{2 \frac{10^{\frac{1}{2} \cdot 2} \tan^{2} (t)}{2^{2}} + {\sqrt{5}}^{2}}} \cdot \frac{\sqrt{10} \sec^{2} (t)}{2} d t$

Step 3.1.1.7.3

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

$3 \int \frac{{(\frac{\sqrt{5}}{\sqrt{2}} \tan (t))}^{2}}{\sqrt{2 \frac{10^{\frac{2}{2}} \tan^{2} (t)}{2^{2}} + {\sqrt{5}}^{2}}} \cdot \frac{\sqrt{10} \sec^{2} (t)}{2} d t$

Step 3.1.1.7.4

Cancel the common factor of $2$ .

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Step 3.1.1.7.4.1

Cancel the common factor.

$3 \int \frac{{(\frac{\sqrt{5}}{\sqrt{2}} \tan (t))}^{2}}{\sqrt{2 \frac{10^{\frac{2}{2}} \tan^{2} (t)}{2^{2}} + {\sqrt{5}}^{2}}} \cdot \frac{\sqrt{10} \sec^{2} (t)}{2} d t$

Step 3.1.1.7.4.2

Rewrite the expression.

$3 \int \frac{{(\frac{\sqrt{5}}{\sqrt{2}} \tan (t))}^{2}}{\sqrt{2 \frac{10^{1} \tan^{2} (t)}{2^{2}} + {\sqrt{5}}^{2}}} \cdot \frac{\sqrt{10} \sec^{2} (t)}{2} d t$

Step 3.1.1.7.5

Evaluate the exponent.

$3 \int \frac{{(\frac{\sqrt{5}}{\sqrt{2}} \tan (t))}^{2}}{\sqrt{2 \frac{10 \tan^{2} (t)}{2^{2}} + {\sqrt{5}}^{2}}} \cdot \frac{\sqrt{10} \sec^{2} (t)}{2} d t$

Step 3.1.1.8

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

$3 \int \frac{{(\frac{\sqrt{5}}{\sqrt{2}} \tan (t))}^{2}}{\sqrt{2 \frac{10 \tan^{2} (t)}{4} + {\sqrt{5}}^{2}}} \cdot \frac{\sqrt{10} \sec^{2} (t)}{2} d t$

Step 3.1.1.9

Cancel the common factor of $2$ .

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Step 3.1.1.9.1

Factor $2$ out of $4$ .

$3 \int \frac{{(\frac{\sqrt{5}}{\sqrt{2}} \tan (t))}^{2}}{\sqrt{2 \frac{10 \tan^{2} (t)}{2 (2)} + {\sqrt{5}}^{2}}} \cdot \frac{\sqrt{10} \sec^{2} (t)}{2} d t$

Step 3.1.1.9.2

Cancel the common factor.

$3 \int \frac{{(\frac{\sqrt{5}}{\sqrt{2}} \tan (t))}^{2}}{\sqrt{2 \frac{10 \tan^{2} (t)}{2 \cdot 2} + {\sqrt{5}}^{2}}} \cdot \frac{\sqrt{10} \sec^{2} (t)}{2} d t$

Step 3.1.1.9.3

Rewrite the expression.

$3 \int \frac{{(\frac{\sqrt{5}}{\sqrt{2}} \tan (t))}^{2}}{\sqrt{\frac{10 \tan^{2} (t)}{2} + {\sqrt{5}}^{2}}} \cdot \frac{\sqrt{10} \sec^{2} (t)}{2} d t$

Step 3.1.1.10

Cancel the common factor of $10$ and $2$ .

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Step 3.1.1.10.1

Factor $2$ out of $10 \tan^{2} (t)$ .

$3 \int \frac{{(\frac{\sqrt{5}}{\sqrt{2}} \tan (t))}^{2}}{\sqrt{\frac{2 (5 \tan^{2} (t))}{2} + {\sqrt{5}}^{2}}} \cdot \frac{\sqrt{10} \sec^{2} (t)}{2} d t$

Step 3.1.1.10.2

Cancel the common factors.

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Step 3.1.1.10.2.1

Factor $2$ out of $2$ .

$3 \int \frac{{(\frac{\sqrt{5}}{\sqrt{2}} \tan (t))}^{2}}{\sqrt{\frac{2 (5 \tan^{2} (t))}{2 (1)} + {\sqrt{5}}^{2}}} \cdot \frac{\sqrt{10} \sec^{2} (t)}{2} d t$

Step 3.1.1.10.2.2

Cancel the common factor.

$3 \int \frac{{(\frac{\sqrt{5}}{\sqrt{2}} \tan (t))}^{2}}{\sqrt{\frac{2 (5 \tan^{2} (t))}{2 \cdot 1} + {\sqrt{5}}^{2}}} \cdot \frac{\sqrt{10} \sec^{2} (t)}{2} d t$

Step 3.1.1.10.2.3

Rewrite the expression.

$3 \int \frac{{(\frac{\sqrt{5}}{\sqrt{2}} \tan (t))}^{2}}{\sqrt{\frac{5 \tan^{2} (t)}{1} + {\sqrt{5}}^{2}}} \cdot \frac{\sqrt{10} \sec^{2} (t)}{2} d t$

Step 3.1.1.10.2.4

Divide $5 \tan^{2} (t)$ by $1$ .

$3 \int \frac{{(\frac{\sqrt{5}}{\sqrt{2}} \tan (t))}^{2}}{\sqrt{5 \tan^{2} (t) + {\sqrt{5}}^{2}}} \cdot \frac{\sqrt{10} \sec^{2} (t)}{2} d t$

Step 3.1.1.11

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 3.1.1.11.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$3 \int \frac{{(\frac{\sqrt{5}}{\sqrt{2}} \tan (t))}^{2}}{\sqrt{5 \tan^{2} (t) + {(5^{\frac{1}{2}})}^{2}}} \cdot \frac{\sqrt{10} \sec^{2} (t)}{2} d t$

Step 3.1.1.11.2

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

$3 \int \frac{{(\frac{\sqrt{5}}{\sqrt{2}} \tan (t))}^{2}}{\sqrt{5 \tan^{2} (t) + 5^{\frac{1}{2} \cdot 2}}} \cdot \frac{\sqrt{10} \sec^{2} (t)}{2} d t$

Step 3.1.1.11.3

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

$3 \int \frac{{(\frac{\sqrt{5}}{\sqrt{2}} \tan (t))}^{2}}{\sqrt{5 \tan^{2} (t) + 5^{\frac{2}{2}}}} \cdot \frac{\sqrt{10} \sec^{2} (t)}{2} d t$

Step 3.1.1.11.4

Cancel the common factor of $2$ .

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Step 3.1.1.11.4.1

Cancel the common factor.

$3 \int \frac{{(\frac{\sqrt{5}}{\sqrt{2}} \tan (t))}^{2}}{\sqrt{5 \tan^{2} (t) + 5^{\frac{2}{2}}}} \cdot \frac{\sqrt{10} \sec^{2} (t)}{2} d t$

Step 3.1.1.11.4.2

Rewrite the expression.

$3 \int \frac{{(\frac{\sqrt{5}}{\sqrt{2}} \tan (t))}^{2}}{\sqrt{5 \tan^{2} (t) + 5^{1}}} \cdot \frac{\sqrt{10} \sec^{2} (t)}{2} d t$

Step 3.1.1.11.5

Evaluate the exponent.

$3 \int \frac{{(\frac{\sqrt{5}}{\sqrt{2}} \tan (t))}^{2}}{\sqrt{5 \tan^{2} (t) + 5}} \cdot \frac{\sqrt{10} \sec^{2} (t)}{2} d t$

Step 3.1.2

Factor $5$ out of $5 \tan^{2} (t) + 5$ .

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Step 3.1.2.1

Factor $5$ out of $5 \tan^{2} (t)$ .

$3 \int \frac{{(\frac{\sqrt{5}}{\sqrt{2}} \tan (t))}^{2}}{\sqrt{5 (\tan^{2} (t)) + 5}} \cdot \frac{\sqrt{10} \sec^{2} (t)}{2} d t$

Step 3.1.2.2

Factor $5$ out of $5$ .

$3 \int \frac{{(\frac{\sqrt{5}}{\sqrt{2}} \tan (t))}^{2}}{\sqrt{5 (\tan^{2} (t)) + 5 (1)}} \cdot \frac{\sqrt{10} \sec^{2} (t)}{2} d t$

Step 3.1.2.3

Factor $5$ out of $5 (\tan^{2} (t)) + 5 (1)$ .

$3 \int \frac{{(\frac{\sqrt{5}}{\sqrt{2}} \tan (t))}^{2}}{\sqrt{5 (\tan^{2} (t) + 1)}} \cdot \frac{\sqrt{10} \sec^{2} (t)}{2} d t$

Step 3.1.3

Apply pythagorean identity.

$3 \int \frac{{(\frac{\sqrt{5}}{\sqrt{2}} \tan (t))}^{2}}{\sqrt{5 \sec^{2} (t)}} \cdot \frac{\sqrt{10} \sec^{2} (t)}{2} d t$

Step 3.1.4

Reorder $5$ and $\sec^{2} (t)$ .

$3 \int \frac{{(\frac{\sqrt{5}}{\sqrt{2}} \tan (t))}^{2}}{\sqrt{\sec^{2} (t) \cdot 5}} \cdot \frac{\sqrt{10} \sec^{2} (t)}{2} d t$

Step 3.1.5

Pull terms out from under the radical.

$3 \int \frac{{(\frac{\sqrt{5}}{\sqrt{2}} \tan (t))}^{2}}{\sec (t) \sqrt{5}} \cdot \frac{\sqrt{10} \sec^{2} (t)}{2} d t$

Step 3.2

Combine fractions.

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Step 3.2.1

Combine $\frac{\sqrt{5}}{\sqrt{2}}$ and $\tan (t)$ .

$3 \int \frac{{(\frac{\sqrt{5} \tan (t)}{\sqrt{2}})}^{2}}{\sec (t) \sqrt{5}} \cdot \frac{\sqrt{10} \sec^{2} (t)}{2} d t$

Step 3.2.2

Simplify the expression.

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Step 3.2.2.1

Apply the product rule to $\frac{\sqrt{5} \tan (t)}{\sqrt{2}}$ .

$3 \int \frac{\frac{{(\sqrt{5} \tan (t))}^{2}}{{\sqrt{2}}^{2}}}{\sec (t) \sqrt{5}} \cdot \frac{\sqrt{10} \sec^{2} (t)}{2} d t$

Step 3.2.2.2

Apply the product rule to $\sqrt{5} \tan (t)$ .

$3 \int \frac{\frac{{\sqrt{5}}^{2} \tan^{2} (t)}{{\sqrt{2}}^{2}}}{\sec (t) \sqrt{5}} \cdot \frac{\sqrt{10} \sec^{2} (t)}{2} d t$

Step 3.2.2.3

Simplify.

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Step 3.2.2.3.1

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 3.2.2.3.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$3 \int \frac{\frac{{(5^{\frac{1}{2}})}^{2} \tan^{2} (t)}{{\sqrt{2}}^{2}}}{\sec (t) \sqrt{5}} \cdot \frac{\sqrt{10} \sec^{2} (t)}{2} d t$

Step 3.2.2.3.1.2

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

$3 \int \frac{\frac{5^{\frac{1}{2} \cdot 2} \tan^{2} (t)}{{\sqrt{2}}^{2}}}{\sec (t) \sqrt{5}} \cdot \frac{\sqrt{10} \sec^{2} (t)}{2} d t$

Step 3.2.2.3.1.3

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

$3 \int \frac{\frac{5^{\frac{2}{2}} \tan^{2} (t)}{{\sqrt{2}}^{2}}}{\sec (t) \sqrt{5}} \cdot \frac{\sqrt{10} \sec^{2} (t)}{2} d t$

Step 3.2.2.3.1.4

Cancel the common factor of $2$ .

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Step 3.2.2.3.1.4.1

Cancel the common factor.

$3 \int \frac{\frac{5^{\frac{2}{2}} \tan^{2} (t)}{{\sqrt{2}}^{2}}}{\sec (t) \sqrt{5}} \cdot \frac{\sqrt{10} \sec^{2} (t)}{2} d t$

Step 3.2.2.3.1.4.2

Rewrite the expression.

$3 \int \frac{\frac{5^{1} \tan^{2} (t)}{{\sqrt{2}}^{2}}}{\sec (t) \sqrt{5}} \cdot \frac{\sqrt{10} \sec^{2} (t)}{2} d t$

Step 3.2.2.3.1.5

Evaluate the exponent.

$3 \int \frac{\frac{5 \tan^{2} (t)}{{\sqrt{2}}^{2}}}{\sec (t) \sqrt{5}} \cdot \frac{\sqrt{10} \sec^{2} (t)}{2} d t$

Step 3.2.2.3.2

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

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Step 3.2.2.3.2.1

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

$3 \int \frac{\frac{5 \tan^{2} (t)}{{(2^{\frac{1}{2}})}^{2}}}{\sec (t) \sqrt{5}} \cdot \frac{\sqrt{10} \sec^{2} (t)}{2} d t$

Step 3.2.2.3.2.2

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

$3 \int \frac{\frac{5 \tan^{2} (t)}{2^{\frac{1}{2} \cdot 2}}}{\sec (t) \sqrt{5}} \cdot \frac{\sqrt{10} \sec^{2} (t)}{2} d t$

Step 3.2.2.3.2.3

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

$3 \int \frac{\frac{5 \tan^{2} (t)}{2^{\frac{2}{2}}}}{\sec (t) \sqrt{5}} \cdot \frac{\sqrt{10} \sec^{2} (t)}{2} d t$

Step 3.2.2.3.2.4

Cancel the common factor of $2$ .

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Step 3.2.2.3.2.4.1

Cancel the common factor.

$3 \int \frac{\frac{5 \tan^{2} (t)}{2^{\frac{2}{2}}}}{\sec (t) \sqrt{5}} \cdot \frac{\sqrt{10} \sec^{2} (t)}{2} d t$

Step 3.2.2.3.2.4.2

Rewrite the expression.

$3 \int \frac{\frac{5 \tan^{2} (t)}{2^{1}}}{\sec (t) \sqrt{5}} \cdot \frac{\sqrt{10} \sec^{2} (t)}{2} d t$

Step 3.2.2.3.2.5

Evaluate the exponent.

$3 \int \frac{\frac{5 \tan^{2} (t)}{2}}{\sec (t) \sqrt{5}} \cdot \frac{\sqrt{10} \sec^{2} (t)}{2} d t$

Step 3.2.2.3.3

Rewrite $\frac{\frac{5 \tan^{2} (t)}{2}}{\sec (t) \sqrt{5}}$ as a product.

$3 \int \frac{5 \tan^{2} (t)}{2} \cdot \frac{1}{\sec (t) \sqrt{5}} \frac{\sqrt{10} \sec^{2} (t)}{2} d t$

Step 3.2.2.3.4

Multiply $\frac{5 \tan^{2} (t)}{2}$ by $\frac{1}{\sec (t) \sqrt{5}}$ .

$3 \int \frac{5 \tan^{2} (t)}{2 (\sec (t) \sqrt{5})} \cdot \frac{\sqrt{10} \sec^{2} (t)}{2} d t$

Step 3.2.2.3.5

Multiply $\frac{5 \tan^{2} (t)}{2 \sec (t) \sqrt{5}}$ by $\frac{\sqrt{10} \sec^{2} (t)}{2}$ .

$3 \int \frac{5 \tan^{2} (t) (\sqrt{10} \sec^{2} (t))}{2 \sec (t) \sqrt{5} \cdot 2} d t$

Step 3.2.2.3.6

Multiply $2$ by $2$ .

$3 \int \frac{5 \tan^{2} (t) (\sqrt{10} \sec^{2} (t))}{4 \sec (t) \sqrt{5}} d t$

Step 3.2.2.3.7

Cancel the common factor of $\sec^{2} (t)$ and $\sec (t)$ .

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Step 3.2.2.3.7.1

Factor $\sec (t)$ out of $5 \tan^{2} (t) \sqrt{10} \sec^{2} (t)$ .

$3 \int \frac{\sec (t) (5 \tan^{2} (t) \sqrt{10} \sec (t))}{4 \sec (t) \sqrt{5}} d t$

Step 3.2.2.3.7.2

Cancel the common factors.

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Step 3.2.2.3.7.2.1

Factor $\sec (t)$ out of $4 \sec (t) \sqrt{5}$ .

$3 \int \frac{\sec (t) (5 \tan^{2} (t) \sqrt{10} \sec (t))}{\sec (t) (4 \sqrt{5})} d t$

Step 3.2.2.3.7.2.2

Cancel the common factor.

$3 \int \frac{\sec (t) (5 \tan^{2} (t) \sqrt{10} \sec (t))}{\sec (t) (4 \sqrt{5})} d t$

Step 3.2.2.3.7.2.3

Rewrite the expression.

$3 \int \frac{5 \tan^{2} (t) \sqrt{10} \sec (t)}{4 \sqrt{5}} d t$

Step 4

Since $\frac{5 \sqrt{10}}{4 \sqrt{5}}$ is constant with respect to $t$ , move $\frac{5 \sqrt{10}}{4 \sqrt{5}}$ out of the integral.

$3 (\frac{5 \sqrt{10}}{4 \sqrt{5}} \int \tan^{2} (t) \sec (t) d t)$

Step 5

Simplify.

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Step 5.1

Combine $\frac{5 \sqrt{10}}{4 \sqrt{5}}$ and $3$ .

$\frac{5 \sqrt{10} \cdot 3}{4 \sqrt{5}} \int \tan^{2} (t) \sec (t) d t$

Step 5.2

Multiply $3$ by $5$ .

$\frac{15 \sqrt{10}}{4 \sqrt{5}} \int \tan^{2} (t) \sec (t) d t$

Step 6

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

$\frac{15 \sqrt{10}}{4 \sqrt{5}} \int \tan^{2} (t) \sec^{1} (t) d t$

Step 7

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

$\frac{15 \sqrt{10}}{4 \sqrt{5}} \int (- 1 + \sec^{2} (t)) \sec^{1} (t) d t$

Step 8

Simplify terms.

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Step 8.1

Apply the distributive property.

$\frac{15 \sqrt{10}}{4 \sqrt{5}} \int - 1 \sec^{1} (t) + \sec^{2} (t) \sec^{1} (t) d t$

Step 8.2

Simplify each term.

$\frac{15 \sqrt{10}}{4 \sqrt{5}} \int - \sec (t) + \sec^{3} (t) d t$

Step 9

Split the single integral into multiple integrals.

$\frac{15 \sqrt{10}}{4 \sqrt{5}} (\int - \sec (t) d t + \int \sec^{3} (t) d t)$

Step 10

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

$\frac{15 \sqrt{10}}{4 \sqrt{5}} (- \int \sec (t) d t + \int \sec^{3} (t) d t)$

Step 11

The integral of $\sec (t)$ with respect to $t$ is $\ln (| \sec (t) + \tan (t) |)$ .

$\frac{15 \sqrt{10}}{4 \sqrt{5}} (- (\ln (| \sec (t) + \tan (t) |) + C) + \int \sec^{3} (t) d t)$

Step 12

Factor $\sec (t)$ out of $\sec^{3} (t)$ .

$\frac{15 \sqrt{10}}{4 \sqrt{5}} (- (\ln (| \sec (t) + \tan (t) |) + C) + \int \sec (t) \sec^{2} (t) d t)$

Step 13

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = \sec (t)$ and $d v = \sec^{2} (t)$ .

$\frac{15 \sqrt{10}}{4 \sqrt{5}} (- (\ln (| \sec (t) + \tan (t) |) + C) + \sec (t) \tan (t) - \int \tan (t) (\sec (t) \tan (t)) d t)$

Step 14

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

$\frac{15 \sqrt{10}}{4 \sqrt{5}} (- (\ln (| \sec (t) + \tan (t) |) + C) + \sec (t) \tan (t) - \int \tan^{1} (t) \tan (t) \sec (t) d t)$

Step 15

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

$\frac{15 \sqrt{10}}{4 \sqrt{5}} (- (\ln (| \sec (t) + \tan (t) |) + C) + \sec (t) \tan (t) - \int \tan^{1} (t) \tan^{1} (t) \sec (t) d t)$

Step 16

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

$\frac{15 \sqrt{10}}{4 \sqrt{5}} (- (\ln (| \sec (t) + \tan (t) |) + C) + \sec (t) \tan (t) - \int {\tan (t)}^{1 + 1} \sec (t) d t)$

Step 17

Simplify the expression.

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Step 17.1

Add $1$ and $1$ .

$\frac{15 \sqrt{10}}{4 \sqrt{5}} (- (\ln (| \sec (t) + \tan (t) |) + C) + \sec (t) \tan (t) - \int \tan^{2} (t) \sec (t) d t)$

Step 17.2

Reorder $\tan^{2} (t)$ and $\sec (t)$ .

$\frac{15 \sqrt{10}}{4 \sqrt{5}} (- (\ln (| \sec (t) + \tan (t) |) + C) + \sec (t) \tan (t) - \int \sec (t) \tan^{2} (t) d t)$

Step 18

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

$\frac{15 \sqrt{10}}{4 \sqrt{5}} (- (\ln (| \sec (t) + \tan (t) |) + C) + \sec (t) \tan (t) - \int \sec (t) (- 1 + \sec^{2} (t)) d t)$

Step 19

Simplify by multiplying through.

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Step 19.1

Rewrite the exponentiation as a product.

$\frac{15 \sqrt{10}}{4 \sqrt{5}} (- (\ln (| \sec (t) + \tan (t) |) + C) + \sec (t) \tan (t) - \int \sec (t) (- 1 + \sec (t) \sec (t)) d t)$

Step 19.2

Apply the distributive property.

$\frac{15 \sqrt{10}}{4 \sqrt{5}} (- (\ln (| \sec (t) + \tan (t) |) + C) + \sec (t) \tan (t) - \int \sec (t) \cdot - 1 + \sec (t) (\sec (t) \sec (t)) d t)$

Step 19.3

Reorder $\sec (t)$ and $- 1$ .

$\frac{15 \sqrt{10}}{4 \sqrt{5}} (- (\ln (| \sec (t) + \tan (t) |) + C) + \sec (t) \tan (t) - \int - 1 \cdot \sec (t) + \sec (t) (\sec (t) \sec (t)) d t)$

Step 20

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

$\frac{15 \sqrt{10}}{4 \sqrt{5}} (- (\ln (| \sec (t) + \tan (t) |) + C) + \sec (t) \tan (t) - \int - 1 \sec (t) + \sec^{1} (t) \sec (t) \sec (t) d t)$

Step 21

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

$\frac{15 \sqrt{10}}{4 \sqrt{5}} (- (\ln (| \sec (t) + \tan (t) |) + C) + \sec (t) \tan (t) - \int - 1 \sec (t) + \sec^{1} (t) \sec^{1} (t) \sec (t) d t)$

Step 22

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

$\frac{15 \sqrt{10}}{4 \sqrt{5}} (- (\ln (| \sec (t) + \tan (t) |) + C) + \sec (t) \tan (t) - \int - 1 \sec (t) + {\sec (t)}^{1 + 1} \sec (t) d t)$

Step 23

Add $1$ and $1$ .

$\frac{15 \sqrt{10}}{4 \sqrt{5}} (- (\ln (| \sec (t) + \tan (t) |) + C) + \sec (t) \tan (t) - \int - 1 \sec (t) + \sec^{2} (t) \sec (t) d t)$

Step 24

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

$\frac{15 \sqrt{10}}{4 \sqrt{5}} (- (\ln (| \sec (t) + \tan (t) |) + C) + \sec (t) \tan (t) - \int - 1 \sec (t) + \sec^{2} (t) \sec^{1} (t) d t)$

Step 25

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

$\frac{15 \sqrt{10}}{4 \sqrt{5}} (- (\ln (| \sec (t) + \tan (t) |) + C) + \sec (t) \tan (t) - \int - 1 \sec (t) + {\sec (t)}^{2 + 1} d t)$

Step 26

Add $2$ and $1$ .

$\frac{15 \sqrt{10}}{4 \sqrt{5}} (- (\ln (| \sec (t) + \tan (t) |) + C) + \sec (t) \tan (t) - \int - 1 \sec (t) + \sec^{3} (t) d t)$

Step 27

Split the single integral into multiple integrals.

$\frac{15 \sqrt{10}}{4 \sqrt{5}} (- (\ln (| \sec (t) + \tan (t) |) + C) + \sec (t) \tan (t) - (\int - 1 \sec (t) d t + \int \sec^{3} (t) d t))$

Step 28

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

$\frac{15 \sqrt{10}}{4 \sqrt{5}} (- (\ln (| \sec (t) + \tan (t) |) + C) + \sec (t) \tan (t) - (- \int \sec (t) d t + \int \sec^{3} (t) d t))$

Step 29

The integral of $\sec (t)$ with respect to $t$ is $\ln (| \sec (t) + \tan (t) |)$ .

$\frac{15 \sqrt{10}}{4 \sqrt{5}} (- (\ln (| \sec (t) + \tan (t) |) + C) + \sec (t) \tan (t) - (- (\ln (| \sec (t) + \tan (t) |) + C) + \int \sec^{3} (t) d t))$

Step 30

Simplify by multiplying through.

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Step 30.1

Apply the distributive property.

$\frac{15 \sqrt{10}}{4 \sqrt{5}} (- (\ln (| \sec (t) + \tan (t) |) + C) + \sec (t) \tan (t) - - (\ln (| \sec (t) + \tan (t) |) + C) - \int \sec^{3} (t) d t)$

Step 30.2

Multiply $- 1$ by $- 1$ .

$\frac{15 \sqrt{10}}{4 \sqrt{5}} (- (\ln (| \sec (t) + \tan (t) |) + C) + \sec (t) \tan (t) + 1 (\ln (| \sec (t) + \tan (t) |) + C) - \int \sec^{3} (t) d t)$

Step 31

Solving for $\int \sec^{3} (t) d t$ , we find that $\int \sec^{3} (t) d t$ = $\frac{\sec (t) \tan (t) + 1 (\ln (| \sec (t) + \tan (t) |) + C)}{2}$ .

$\frac{15 \sqrt{10}}{4 \sqrt{5}} (- (\ln (| \sec (t) + \tan (t) |) + C) + \frac{\sec (t) \tan (t) + 1 (\ln (| \sec (t) + \tan (t) |) + C)}{2} + C)$

Step 32

Multiply $\ln (| \sec (t) + \tan (t) |) + C$ by $1$ .

$\frac{15 \sqrt{10}}{4 \sqrt{5}} (- (\ln (| \sec (t) + \tan (t) |) + C) + \frac{\sec (t) \tan (t) + \ln (| \sec (t) + \tan (t) |) + C}{2} + C)$

Step 33

Simplify.

$\frac{15 \sqrt{\frac{10}{5}}}{4} (- \ln (| \sec (t) + \tan (t) |) + \frac{\sec (t) \tan (t) + \ln (| \sec (t) + \tan (t) |)}{2}) + C$

Step 34

Simplify.

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Step 34.1

Divide $10$ by $5$ .

$\frac{15 \sqrt{2}}{4} (- \ln (| \sec (t) + \tan (t) |) + \frac{\sec (t) \tan (t) + \ln (| \sec (t) + \tan (t) |)}{2}) + C$

Step 34.2

To write $- \ln (| \sec (t) + \tan (t) |)$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{15 \sqrt{2}}{4} (- \ln (| \sec (t) + \tan (t) |) \cdot \frac{2}{2} + \frac{\sec (t) \tan (t) + \ln (| \sec (t) + \tan (t) |)}{2}) + C$

Step 34.3

Combine $- \ln (| \sec (t) + \tan (t) |)$ and $\frac{2}{2}$ .

$\frac{15 \sqrt{2}}{4} (\frac{- \ln (| \sec (t) + \tan (t) |) \cdot 2}{2} + \frac{\sec (t) \tan (t) + \ln (| \sec (t) + \tan (t) |)}{2}) + C$

Step 34.4

Combine the numerators over the common denominator.

$\frac{15 \sqrt{2}}{4} \cdot \frac{- \ln (| \sec (t) + \tan (t) |) \cdot 2 + \sec (t) \tan (t) + \ln (| \sec (t) + \tan (t) |)}{2} + C$

Step 34.5

Multiply $2$ by $- 1$ .

$\frac{15 \sqrt{2}}{4} \cdot \frac{- 2 \ln (| \sec (t) + \tan (t) |) + \sec (t) \tan (t) + \ln (| \sec (t) + \tan (t) |)}{2} + C$

Step 34.6

Add $- 2 \ln (| \sec (t) + \tan (t) |)$ and $\ln (| \sec (t) + \tan (t) |)$ .

$\frac{15 \sqrt{2}}{4} \cdot \frac{\sec (t) \tan (t) - \ln (| \sec (t) + \tan (t) |)}{2} + C$

Step 34.7

Multiply $\frac{15 \sqrt{2}}{4}$ by $\frac{\sec (t) \tan (t) - \ln (| \sec (t) + \tan (t) |)}{2}$ .

$\frac{15 \sqrt{2} (\sec (t) \tan (t) - \ln (| \sec (t) + \tan (t) |))}{4 \cdot 2} + C$

Step 34.8

Multiply $4$ by $2$ .

$\frac{15 \sqrt{2} (\sec (t) \tan (t) - \ln (| \sec (t) + \tan (t) |))}{8} + C$

Step 35

Replace all occurrences of $t$ with $\arctan (\frac{\sqrt{2} x}{\sqrt{5}})$ .

$\frac{15 \sqrt{2} (\sec (\arctan (\frac{\sqrt{2} x}{\sqrt{5}})) \tan (\arctan (\frac{\sqrt{2} x}{\sqrt{5}})) - \ln (| \sec (\arctan (\frac{\sqrt{2} x}{\sqrt{5}})) + \tan (\arctan (\frac{\sqrt{2} x}{\sqrt{5}})) |))}{8} + C$

Step 36

Reorder terms.

$\frac{15 \sqrt{2}}{8} (\sec (\arctan (\frac{\sqrt{2}}{\sqrt{5}} x)) \tan (\arctan (\frac{\sqrt{2}}{\sqrt{5}} x)) - \ln (| \sec (\arctan (\frac{\sqrt{2}}{\sqrt{5}} x)) + \tan (\arctan (\frac{\sqrt{2}}{\sqrt{5}} x)) |)) + C$