Calculus Examples

$\int x^{3} \sqrt{4 + x^{2}} d x$

Step 1

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

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

Step 2

Simplify terms.

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Step 2.1

Simplify $\sqrt{4 + {(2 \tan (t))}^{2}}$ .

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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 $2 \tan (t)$ .

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

Step 2.1.1.2

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

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

Step 2.1.2

Factor $4$ out of $4$ .

$\int {(2 \tan (t))}^{3} \sqrt{4 (1) + 4 \tan^{2} (t)} (2 \sec^{2} (t)) d t$

Step 2.1.3

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

$\int {(2 \tan (t))}^{3} \sqrt{4 (1) + 4 (\tan^{2} (t))} (2 \sec^{2} (t)) d t$

Step 2.1.4

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

$\int {(2 \tan (t))}^{3} \sqrt{4 (1 + \tan^{2} (t))} (2 \sec^{2} (t)) d t$

Step 2.1.5

Rearrange terms.

$\int {(2 \tan (t))}^{3} \sqrt{4 (\tan^{2} (t) + 1)} (2 \sec^{2} (t)) d t$

Step 2.1.6

Apply pythagorean identity.

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

Step 2.1.7

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

$\int {(2 \tan (t))}^{3} \sqrt{{(2 \sec (t))}^{2}} (2 \sec^{2} (t)) d t$

Step 2.1.8

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

$\int {(2 \tan (t))}^{3} (2 \sec (t)) (2 \sec^{2} (t)) d t$

Step 2.2

Simplify.

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Step 2.2.1

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

$\int {(2 (\tan (t)))}^{3} (2 \sec (t)) (2 \sec^{2} (t)) d t$

Step 2.2.2

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

$\int 2^{3} \tan^{3} (t) (2 \sec (t)) (2 \sec^{2} (t)) d t$

Step 2.2.3

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

$\int 8 \tan^{3} (t) (2 \sec (t)) (2 \sec^{2} (t)) d t$

Step 2.2.4

Multiply $2$ by $8$ .

$\int 16 \tan^{3} (t) \sec (t) (2 \sec^{2} (t)) d t$

Step 2.2.5

Multiply $2$ by $16$ .

$\int 32 \tan^{3} (t) \sec (t) \sec^{2} (t) d t$

Step 2.2.6

Multiply $\sec (t)$ by $\sec^{2} (t)$ by adding the exponents.

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Step 2.2.6.1

Move $\sec^{2} (t)$ .

$\int 32 \tan^{3} (t) (\sec^{2} (t) \sec (t)) d t$

Step 2.2.6.2

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

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Step 2.2.6.2.1

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

$\int 32 \tan^{3} (t) (\sec^{2} (t) \sec^{1} (t)) d t$

Step 2.2.6.2.2

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

$\int 32 \tan^{3} (t) {\sec (t)}^{2 + 1} d t$

Step 2.2.6.3

Add $2$ and $1$ .

$\int 32 \tan^{3} (t) \sec^{3} (t) d t$

Step 3

Since $32$ is constant with respect to $t$ , move $32$ out of the integral.

$32 \int \tan^{3} (t) \sec^{3} (t) d t$

Step 4

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

$32 \int \tan^{2} (t) \tan (t) \sec^{3} (t) d t$

Step 5

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

$32 \int (- 1 + \sec^{2} (t)) \tan (t) \sec^{3} (t) d t$

Step 6

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

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Step 6.1

Let $u = \sec (t)$ . Find $\frac{d u}{d t}$ .

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Step 6.1.1

Differentiate $\sec (t)$ .

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

Step 6.1.2

The derivative of $\sec (t)$ with respect to $t$ is $\sec (t) \tan (t)$ .

$\sec (t) \tan (t)$

Step 6.2

Rewrite the problem using $u$ and $d u$ .

$32 \int (- 1 + u^{2}) u^{2} d u$

Step 7

Multiply $(- 1 + u^{2}) u^{2}$ .

$32 \int - 1 u^{2} + u^{2} u^{2} d u$

Step 8

Simplify.

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Step 8.1

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

$32 \int - u^{2} + u^{2} u^{2} d u$

Step 8.2

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

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Step 8.2.1

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

$32 \int - u^{2} + u^{2 + 2} d u$

Step 8.2.2

Add $2$ and $2$ .

$32 \int - u^{2} + u^{4} d u$

Step 9

Split the single integral into multiple integrals.

$32 (\int - u^{2} d u + \int u^{4} d u)$

Step 10

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

$32 (- \int u^{2} d u + \int u^{4} d u)$

Step 11

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

$32 (- (\frac{1}{3} u^{3} + C) + \int u^{4} d u)$

Step 12

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

$32 (- (\frac{1}{3} u^{3} + C) + \frac{1}{5} u^{5} + C)$

Step 13

Simplify.

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Step 13.1

Simplify.

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Step 13.1.1

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

$32 (- (\frac{u^{3}}{3} + C) + \frac{1}{5} u^{5} + C)$

Step 13.1.2

Combine $\frac{1}{5}$ and $u^{5}$ .

$32 (- (\frac{u^{3}}{3} + C) + \frac{u^{5}}{5} + C)$

Step 13.2

Simplify.

$32 (- \frac{1}{3} u^{3} + \frac{1}{5} u^{5}) + C$

Step 14

Substitute back in for each integration substitution variable.

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Step 14.1

Replace all occurrences of $u$ with $\sec (t)$ .

$32 (- \frac{1}{3} \sec^{3} (t) + \frac{1}{5} \sec^{5} (t)) + C$

Step 14.2

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

$32 (- \frac{1}{3} \sec^{3} (\arctan (\frac{x}{2})) + \frac{1}{5} \sec^{5} (\arctan (\frac{x}{2}))) + C$

Step 15

Reorder terms.

$32 (- \frac{1}{3} \sec^{3} (\arctan (\frac{1}{2} x)) + \frac{1}{5} \sec^{5} (\arctan (\frac{1}{2} x))) + C$