Calculus Examples

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

Step 1

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

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

Step 2

Simplify terms.

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Step 2.1

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

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Step 2.1.1

Simplify each term.

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Step 2.1.1.1

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

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

Step 2.1.1.2

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

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

Step 2.1.2

Factor $4$ out of $4 \sec^{2} (t)$ .

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

Step 2.1.3

Factor $4$ out of $- 4$ .

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

Step 2.1.4

Factor $4$ out of $4 \sec^{2} (t) + 4 \cdot - 1$ .

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

Step 2.1.5

Apply pythagorean identity.

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

Step 2.1.6

Rewrite $4 \tan^{2} (t)$ as ${(2 \tan (t))}^{2}$ .

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

Step 2.1.7

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

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

Step 2.2

Cancel the common factor of $2 \sec (t)$ .

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Step 2.2.1

Factor $2 \sec (t)$ out of $2 \sec (t) \tan (t)$ .

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

Step 2.2.2

Cancel the common factor.

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

Step 2.2.3

Rewrite the expression.

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

Add $1$ and $1$ .

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

Step 7

Since $2$ is constant with respect to $t$ , move $2$ out of the integral.

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

Step 8

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

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

Step 9

Split the single integral into multiple integrals.

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

Step 10

Apply the constant rule.

$2 (- t]_{0}^{\frac{π}{3}} + \int_{0}^{\frac{π}{3}} \sec^{2} (t) d t)$

Step 11

Since the derivative of $\tan (t)$ is $\sec^{2} (t)$ , the integral of $\sec^{2} (t)$ is $\tan (t)$ .

$2 (- t]_{0}^{\frac{π}{3}} + \tan (t)]_{0}^{\frac{π}{3}})$

Step 12

Combine $- t]_{0}^{\frac{π}{3}}$ and $\tan (t)]_{0}^{\frac{π}{3}}$ .

$2 (- t + \tan (t)]_{0}^{\frac{π}{3}})$

Step 13

Substitute and simplify.

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Step 13.1

Evaluate $- t + \tan (t)$ at $\frac{π}{3}$ and at $0$ .

$2 ((- \frac{π}{3} + \tan (\frac{π}{3})) - (- 0 + \tan (0)))$

Step 13.2

Simplify.

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Step 13.2.1

Multiply $- 1$ by $0$ .

$2 (- \frac{π}{3} + \tan (\frac{π}{3}) - (0 + \tan (0)))$

Step 13.2.2

Add $0$ and $\tan (0)$ .

$2 (- \frac{π}{3} + \tan (\frac{π}{3}) - \tan (0))$

Step 14

Simplify.

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Step 14.1

The exact value of $\tan (\frac{π}{3})$ is $\sqrt{3}$ .

$2 (- \frac{π}{3} + \sqrt{3} - \tan (0))$

Step 14.2

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

$2 (- \frac{π}{3} + \sqrt{3} - 0)$

Step 14.3

Multiply $- 1$ by $0$ .

$2 (- \frac{π}{3} + \sqrt{3} + 0)$

Step 14.4

Add $- \frac{π}{3} + \sqrt{3}$ and $0$ .

$2 (- \frac{π}{3} + \sqrt{3})$

Step 15

Simplify.

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Step 15.1

Apply the distributive property.

$2 (- \frac{π}{3}) + 2 \sqrt{3}$

Step 15.2

Multiply $2 (- \frac{π}{3})$ .

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Step 15.2.1

Multiply $- 1$ by $2$ .

$- 2 \frac{π}{3} + 2 \sqrt{3}$

Step 15.2.2

Combine $- 2$ and $\frac{π}{3}$ .

$\frac{- 2 π}{3} + 2 \sqrt{3}$

Step 15.3

Move the negative in front of the fraction.

$- \frac{2 π}{3} + 2 \sqrt{3}$

Step 16

The result can be shown in multiple forms.

Exact Form:

$- \frac{2 π}{3} + 2 \sqrt{3}$

Decimal Form:

$1.36970651 \dots$

Step 17