Calculus Examples

$\int_{1}^{4} \frac{1}{x \sqrt{16 x^{2} - 5}} d x$

Step 1

Let $x = \frac{\sqrt{5}}{4} \sec (t)$ , where $- \frac{π}{2} \leq t \leq \frac{π}{2}$ . Then $d x = \frac{\sqrt{5} \sec (t) \tan (t)}{4} d t$ . Note that since $- \frac{π}{2} \leq t \leq \frac{π}{2}$ , $\frac{\sqrt{5} \sec (t) \tan (t)}{4}$ is positive.

$\int_{0.97759655}^{1.4305831} \frac{1}{\frac{\sqrt{5}}{4} \sec (t) \sqrt{16 {(\frac{\sqrt{5}}{4} \sec (t))}^{2} - 5}} \cdot \frac{\sqrt{5} \sec (t) \tan (t)}{4} d t$

Step 2

Simplify terms.

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Step 2.1

Simplify $\sqrt{16 {(\frac{\sqrt{5}}{4} \sec (t))}^{2} - 5}$ .

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Step 2.1.1

Simplify each term.

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Step 2.1.1.1

Combine $\frac{\sqrt{5}}{4}$ and $\sec (t)$ .

$\int_{0.97759655}^{1.4305831} \frac{1}{\frac{\sqrt{5}}{4} \sec (t) \sqrt{16 {(\frac{\sqrt{5} \sec (t)}{4})}^{2} - 5}} \cdot \frac{\sqrt{5} \sec (t) \tan (t)}{4} d t$

Step 2.1.1.2

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

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Step 2.1.1.2.1

Apply the product rule to $\frac{\sqrt{5} \sec (t)}{4}$ .

$\int_{0.97759655}^{1.4305831} \frac{1}{\frac{\sqrt{5}}{4} \sec (t) \sqrt{16 \frac{{(\sqrt{5} \sec (t))}^{2}}{4^{2}} - 5}} \cdot \frac{\sqrt{5} \sec (t) \tan (t)}{4} d t$

Step 2.1.1.2.2

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

$\int_{0.97759655}^{1.4305831} \frac{1}{\frac{\sqrt{5}}{4} \sec (t) \sqrt{16 \frac{{\sqrt{5}}^{2} \sec^{2} (t)}{4^{2}} - 5}} \cdot \frac{\sqrt{5} \sec (t) \tan (t)}{4} d t$

Step 2.1.1.3

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

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Step 2.1.1.3.1

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

$\int_{0.97759655}^{1.4305831} \frac{1}{\frac{\sqrt{5}}{4} \sec (t) \sqrt{16 \frac{{(5^{\frac{1}{2}})}^{2} \sec^{2} (t)}{4^{2}} - 5}} \cdot \frac{\sqrt{5} \sec (t) \tan (t)}{4} d t$

Step 2.1.1.3.2

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

$\int_{0.97759655}^{1.4305831} \frac{1}{\frac{\sqrt{5}}{4} \sec (t) \sqrt{16 \frac{5^{\frac{1}{2} \cdot 2} \sec^{2} (t)}{4^{2}} - 5}} \cdot \frac{\sqrt{5} \sec (t) \tan (t)}{4} d t$

Step 2.1.1.3.3

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

$\int_{0.97759655}^{1.4305831} \frac{1}{\frac{\sqrt{5}}{4} \sec (t) \sqrt{16 \frac{5^{\frac{2}{2}} \sec^{2} (t)}{4^{2}} - 5}} \cdot \frac{\sqrt{5} \sec (t) \tan (t)}{4} d t$

Step 2.1.1.3.4

Cancel the common factor of $2$ .

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Step 2.1.1.3.4.1

Cancel the common factor.

$\int_{0.97759655}^{1.4305831} \frac{1}{\frac{\sqrt{5}}{4} \sec (t) \sqrt{16 \frac{5^{\frac{2}{2}} \sec^{2} (t)}{4^{2}} - 5}} \cdot \frac{\sqrt{5} \sec (t) \tan (t)}{4} d t$

Step 2.1.1.3.4.2

Rewrite the expression.

$\int_{0.97759655}^{1.4305831} \frac{1}{\frac{\sqrt{5}}{4} \sec (t) \sqrt{16 \frac{5^{1} \sec^{2} (t)}{4^{2}} - 5}} \cdot \frac{\sqrt{5} \sec (t) \tan (t)}{4} d t$

Step 2.1.1.3.5

Evaluate the exponent.

$\int_{0.97759655}^{1.4305831} \frac{1}{\frac{\sqrt{5}}{4} \sec (t) \sqrt{16 \frac{5 \sec^{2} (t)}{4^{2}} - 5}} \cdot \frac{\sqrt{5} \sec (t) \tan (t)}{4} d t$

Step 2.1.1.4

Raise $4$ to the power of $2$ .

$\int_{0.97759655}^{1.4305831} \frac{1}{\frac{\sqrt{5}}{4} \sec (t) \sqrt{16 \frac{5 \sec^{2} (t)}{16} - 5}} \cdot \frac{\sqrt{5} \sec (t) \tan (t)}{4} d t$

Step 2.1.1.5

Cancel the common factor of $16$ .

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Step 2.1.1.5.1

Cancel the common factor.

$\int_{0.97759655}^{1.4305831} \frac{1}{\frac{\sqrt{5}}{4} \sec (t) \sqrt{16 \frac{5 \sec^{2} (t)}{16} - 5}} \cdot \frac{\sqrt{5} \sec (t) \tan (t)}{4} d t$

Step 2.1.1.5.2

Rewrite the expression.

$\int_{0.97759655}^{1.4305831} \frac{1}{\frac{\sqrt{5}}{4} \sec (t) \sqrt{5 \sec^{2} (t) - 5}} \cdot \frac{\sqrt{5} \sec (t) \tan (t)}{4} d t$

Step 2.1.2

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

$\int_{0.97759655}^{1.4305831} \frac{1}{\frac{\sqrt{5}}{4} \sec (t) \sqrt{5 (\sec^{2} (t)) - 5}} \cdot \frac{\sqrt{5} \sec (t) \tan (t)}{4} d t$

Step 2.1.3

Factor $5$ out of $- 5$ .

$\int_{0.97759655}^{1.4305831} \frac{1}{\frac{\sqrt{5}}{4} \sec (t) \sqrt{5 \sec^{2} (t) + 5 \cdot - 1}} \cdot \frac{\sqrt{5} \sec (t) \tan (t)}{4} d t$

Step 2.1.4

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

$\int_{0.97759655}^{1.4305831} \frac{1}{\frac{\sqrt{5}}{4} \sec (t) \sqrt{5 (\sec^{2} (t) - 1)}} \cdot \frac{\sqrt{5} \sec (t) \tan (t)}{4} d t$

Step 2.1.5

Apply pythagorean identity.

$\int_{0.97759655}^{1.4305831} \frac{1}{\frac{\sqrt{5}}{4} \sec (t) \sqrt{5 \tan^{2} (t)}} \cdot \frac{\sqrt{5} \sec (t) \tan (t)}{4} d t$

Step 2.1.6

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

$\int_{0.97759655}^{1.4305831} \frac{1}{\frac{\sqrt{5}}{4} \sec (t) \sqrt{\tan^{2} (t) \cdot 5}} \cdot \frac{\sqrt{5} \sec (t) \tan (t)}{4} d t$

Step 2.1.7

Pull terms out from under the radical.

$\int_{0.97759655}^{1.4305831} \frac{1}{\frac{\sqrt{5}}{4} \sec (t) (\tan (t) \sqrt{5})} \cdot \frac{\sqrt{5} \sec (t) \tan (t)}{4} d t$

Step 2.2

Simplify.

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Step 2.2.1

Combine $\frac{\sqrt{5}}{4}$ and $\sec (t)$ .

$\int_{0.97759655}^{1.4305831} \frac{1}{\frac{\sqrt{5} \sec (t)}{4} (\tan (t) \sqrt{5})} \cdot \frac{\sqrt{5} \sec (t) \tan (t)}{4} d t$

Step 2.2.2

Combine $\tan (t)$ and $\frac{\sqrt{5} \sec (t)}{4}$ .

$\int_{0.97759655}^{1.4305831} \frac{1}{\frac{\tan (t) (\sqrt{5} \sec (t))}{4} \sqrt{5}} \cdot \frac{\sqrt{5} \sec (t) \tan (t)}{4} d t$

Step 2.2.3

Combine $\frac{\tan (t) (\sqrt{5} \sec (t))}{4}$ and $\sqrt{5}$ .

$\int_{0.97759655}^{1.4305831} \frac{1}{\frac{\tan (t) (\sqrt{5} \sec (t)) \sqrt{5}}{4}} \cdot \frac{\sqrt{5} \sec (t) \tan (t)}{4} d t$

Step 2.2.4

Raise $\sqrt{5}$ to the power of $1$ .

$\int_{0.97759655}^{1.4305831} \frac{1}{\frac{\tan (t) ({\sqrt{5}}^{1} \sqrt{5} \sec (t))}{4}} \cdot \frac{\sqrt{5} \sec (t) \tan (t)}{4} d t$

Step 2.2.5

Raise $\sqrt{5}$ to the power of $1$ .

$\int_{0.97759655}^{1.4305831} \frac{1}{\frac{\tan (t) ({\sqrt{5}}^{1} {\sqrt{5}}^{1} \sec (t))}{4}} \cdot \frac{\sqrt{5} \sec (t) \tan (t)}{4} d t$

Step 2.2.6

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

$\int_{0.97759655}^{1.4305831} \frac{1}{\frac{\tan (t) ({\sqrt{5}}^{1 + 1} \sec (t))}{4}} \cdot \frac{\sqrt{5} \sec (t) \tan (t)}{4} d t$

Step 2.2.7

Add $1$ and $1$ .

$\int_{0.97759655}^{1.4305831} \frac{1}{\frac{\tan (t) ({\sqrt{5}}^{2} \sec (t))}{4}} \cdot \frac{\sqrt{5} \sec (t) \tan (t)}{4} d t$

Step 2.2.8

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

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Step 2.2.8.1

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

$\int_{0.97759655}^{1.4305831} \frac{1}{\frac{\tan (t) ({(5^{\frac{1}{2}})}^{2} \sec (t))}{4}} \cdot \frac{\sqrt{5} \sec (t) \tan (t)}{4} d t$

Step 2.2.8.2

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

$\int_{0.97759655}^{1.4305831} \frac{1}{\frac{\tan (t) (5^{\frac{1}{2} \cdot 2} \sec (t))}{4}} \cdot \frac{\sqrt{5} \sec (t) \tan (t)}{4} d t$

Step 2.2.8.3

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

$\int_{0.97759655}^{1.4305831} \frac{1}{\frac{\tan (t) (5^{\frac{2}{2}} \sec (t))}{4}} \cdot \frac{\sqrt{5} \sec (t) \tan (t)}{4} d t$

Step 2.2.8.4

Cancel the common factor of $2$ .

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Step 2.2.8.4.1

Cancel the common factor.

$\int_{0.97759655}^{1.4305831} \frac{1}{\frac{\tan (t) (5^{\frac{2}{2}} \sec (t))}{4}} \cdot \frac{\sqrt{5} \sec (t) \tan (t)}{4} d t$

Step 2.2.8.4.2

Rewrite the expression.

$\int_{0.97759655}^{1.4305831} \frac{1}{\frac{\tan (t) (5^{1} \sec (t))}{4}} \cdot \frac{\sqrt{5} \sec (t) \tan (t)}{4} d t$

Step 2.2.8.5

Evaluate the exponent.

$\int_{0.97759655}^{1.4305831} \frac{1}{\frac{\tan (t) (5 \sec (t))}{4}} \cdot \frac{\sqrt{5} \sec (t) \tan (t)}{4} d t$

Step 2.2.9

Move $5$ to the left of $\tan (t)$ .

$\int_{0.97759655}^{1.4305831} \frac{1}{\frac{5 \cdot \tan (t) \sec (t)}{4}} \cdot \frac{\sqrt{5} \sec (t) \tan (t)}{4} d t$

Step 2.2.10

Multiply by the reciprocal of the fraction to divide by $\frac{5 \tan (t) \sec (t)}{4}$ .

$\int_{0.97759655}^{1.4305831} 1 \frac{4}{5 \tan (t) \sec (t)} \frac{\sqrt{5} \sec (t) \tan (t)}{4} d t$

Step 2.2.11

Multiply $\frac{4}{5 \tan (t) \sec (t)}$ by $1$ .

$\int_{0.97759655}^{1.4305831} \frac{4}{5 \tan (t) \sec (t)} \cdot \frac{\sqrt{5} \sec (t) \tan (t)}{4} d t$

Step 2.2.12

Multiply $\frac{4}{5 \tan (t) \sec (t)}$ by $\frac{\sqrt{5} \sec (t) \tan (t)}{4}$ .

$\int_{0.97759655}^{1.4305831} \frac{4 (\sqrt{5} \sec (t) \tan (t))}{5 \tan (t) \sec (t) \cdot 4} d t$

Step 2.2.13

Multiply $4$ by $5$ .

$\int_{0.97759655}^{1.4305831} \frac{4 (\sqrt{5} \sec (t) \tan (t))}{20 \tan (t) \sec (t)} d t$

Step 2.2.14

Cancel the common factors.

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Step 2.2.14.1

Factor $4$ out of $20 \tan (t) \sec (t)$ .

$\int_{0.97759655}^{1.4305831} \frac{4 (\sqrt{5} \sec (t) \tan (t))}{4 (5 \tan (t) \sec (t))} d t$

Step 2.2.14.2

Cancel the common factor.

$\int_{0.97759655}^{1.4305831} \frac{4 (\sqrt{5} \sec (t) \tan (t))}{4 (5 \tan (t) \sec (t))} d t$

Step 2.2.14.3

Rewrite the expression.

$\int_{0.97759655}^{1.4305831} \frac{\sqrt{5} \sec (t) \tan (t)}{5 \tan (t) \sec (t)} d t$

Step 2.2.15

Cancel the common factor of $\sec (t)$ .

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Step 2.2.15.1

Cancel the common factor.

$\int_{0.97759655}^{1.4305831} \frac{\sqrt{5} \sec (t) \tan (t)}{5 \tan (t) \sec (t)} d t$

Step 2.2.15.2

Rewrite the expression.

$\int_{0.97759655}^{1.4305831} \frac{\sqrt{5} \tan (t)}{5 \tan (t)} d t$

Step 2.2.16

Cancel the common factor of $\tan (t)$ .

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Step 2.2.16.1

Cancel the common factor.

$\int_{0.97759655}^{1.4305831} \frac{\sqrt{5} \tan (t)}{5 \tan (t)} d t$

Step 2.2.16.2

Rewrite the expression.

$\int_{0.97759655}^{1.4305831} \frac{\sqrt{5}}{5} d t$

Step 3

Apply the constant rule.

$\frac{\sqrt{5}}{5} t]_{0.97759655}^{1.4305831}$

Step 4

Substitute and simplify.

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Step 4.1

Evaluate $\frac{\sqrt{5}}{5} t$ at $1.4305831$ and at $0.97759655$ .

$(\frac{\sqrt{5}}{5} \cdot 1.4305831) - \frac{\sqrt{5}}{5} \cdot 0.97759655$

Step 4.2

Simplify.

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Step 4.2.1

Combine $\frac{\sqrt{5}}{5}$ and $1.4305831$ .

$\frac{\sqrt{5} \cdot 1.4305831}{5} - \frac{\sqrt{5}}{5} \cdot 0.97759655$

Step 4.2.2

Multiply $\sqrt{5}$ by $1.4305831$ .

$\frac{3.19888106}{5} - \frac{\sqrt{5}}{5} \cdot 0.97759655$

Step 4.2.3

Multiply $0.97759655$ by $- 1$ .

$\frac{3.19888106}{5} - 0.97759655 \frac{\sqrt{5}}{5}$

Step 4.2.4

Combine $- 0.97759655$ and $\frac{\sqrt{5}}{5}$ .

$\frac{3.19888106}{5} + \frac{- 0.97759655 \sqrt{5}}{5}$

Step 4.2.5

Multiply $- 0.97759655$ by $\sqrt{5}$ .

$\frac{3.19888106}{5} + \frac{- 2.18597234}{5}$

Step 4.2.6

Move the negative in front of the fraction.

$\frac{3.19888106}{5} - \frac{2.18597234}{5}$

Step 4.2.7

Combine the numerators over the common denominator.

$\frac{3.19888106 - 2.18597234}{5}$

Step 4.2.8

Subtract $2.18597234$ from $3.19888106$ .

$\frac{1.01290872}{5}$

Step 5

Divide $1.01290872$ by $5$ .

$0.20258174$

Step 6