Calculus Examples

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

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.

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

Step 2

Simplify terms.

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Step 2.1

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

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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 $2 \tan (t)$ .

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

Step 2.1.1.2

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

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

Step 2.1.2

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

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Step 2.1.2.1

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

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

Step 2.1.2.2

Factor $4$ out of $4$ .

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

Step 2.1.2.3

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

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

Step 2.1.3

Apply pythagorean identity.

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

Step 2.1.4

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

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

Step 2.1.5

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

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

Step 2.2

Reduce the expression by cancelling the common factors.

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Step 2.2.1

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

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Step 2.2.1.1

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

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

Step 2.2.1.2

Cancel the common factor.

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

Step 2.2.1.3

Rewrite the expression.

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

Step 2.2.2

Simplify.

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Step 2.2.2.1

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

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

Step 2.2.2.2

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

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

Step 2.2.2.3

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

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

Step 3

Since $8$ is constant with respect to $t$ , move $8$ out of the integral.

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

Step 4

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

$8 \int \tan^{3} (t) \sec^{1} (t) d t$

Step 5

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

$8 \int \tan^{2} (t) \tan (t) \sec^{1} (t) d t$

Step 6

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

$8 \int (- 1 + \sec^{2} (t)) \tan (t) \sec^{1} (t) d t$

Step 7

Simplify.

$8 \int (- 1 + \sec^{2} (t)) \tan (t) \sec (t) d t$

Step 8

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 8.1

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

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Step 8.1.1

Differentiate $\sec (t)$ .

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

Step 8.1.2

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

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

Step 8.2

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

$8 \int - 1 + u^{2} d u$

Step 9

Split the single integral into multiple integrals.

$8 (\int - 1 d u + \int u^{2} d u)$

Step 10

Apply the constant rule.

$8 (- u + C + \int u^{2} d u)$

Step 11

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

$8 (- u + C + \frac{1}{3} u^{3} + C)$

Step 12

Simplify.

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Step 12.1

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

$8 (- u + C + \frac{u^{3}}{3} + C)$

Step 12.2

Simplify.

$8 (- u + \frac{1}{3} u^{3}) + C$

Step 13

Substitute back in for each integration substitution variable.

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Step 13.1

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

$8 (- \sec (t) + \frac{1}{3} \sec^{3} (t)) + C$

Step 13.2

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

$8 (- \sec (\arctan (\frac{x}{2})) + \frac{1}{3} \sec^{3} (\arctan (\frac{x}{2}))) + C$

Step 14

Simplify.

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Step 14.1

Simplify each term.

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Step 14.1.1

Draw a triangle in the plane with vertices $(1, \frac{x}{2})$ , $(1, 0)$ , and the origin. Then $\arctan (\frac{x}{2})$ is the angle between the positive x-axis and the ray beginning at the origin and passing through $(1, \frac{x}{2})$ . Therefore, $\sec (\arctan (\frac{x}{2}))$ is $\sqrt{1 + {(\frac{x}{2})}^{2}}$ .

$8 (- \sqrt{1 + {(\frac{x}{2})}^{2}} + \frac{1}{3} \sec^{3} (\arctan (\frac{x}{2}))) + C$

Step 14.1.2

Apply the product rule to $\frac{x}{2}$ .

$8 (- \sqrt{1 + \frac{x^{2}}{2^{2}}} + \frac{1}{3} \sec^{3} (\arctan (\frac{x}{2}))) + C$

Step 14.1.3

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

$8 (- \sqrt{1 + \frac{x^{2}}{4}} + \frac{1}{3} \sec^{3} (\arctan (\frac{x}{2}))) + C$

Step 14.1.4

Write $1$ as a fraction with a common denominator.

$8 (- \sqrt{\frac{4}{4} + \frac{x^{2}}{4}} + \frac{1}{3} \sec^{3} (\arctan (\frac{x}{2}))) + C$

Step 14.1.5

Combine the numerators over the common denominator.

$8 (- \sqrt{\frac{4 + x^{2}}{4}} + \frac{1}{3} \sec^{3} (\arctan (\frac{x}{2}))) + C$

Step 14.1.6

Rewrite $\frac{4 + x^{2}}{4}$ as ${(\frac{1}{2})}^{2} (4 + x^{2})$ .

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Step 14.1.6.1

Factor the perfect power $1^{2}$ out of $4 + x^{2}$ .

$8 (- \sqrt{\frac{1^{2} (4 + x^{2})}{4}} + \frac{1}{3} \sec^{3} (\arctan (\frac{x}{2}))) + C$

Step 14.1.6.2

Factor the perfect power $2^{2}$ out of $4$ .

$8 (- \sqrt{\frac{1^{2} (4 + x^{2})}{2^{2} \cdot 1}} + \frac{1}{3} \sec^{3} (\arctan (\frac{x}{2}))) + C$

Step 14.1.6.3

Rearrange the fraction $\frac{1^{2} (4 + x^{2})}{2^{2} \cdot 1}$ .

$8 (- \sqrt{{(\frac{1}{2})}^{2} (4 + x^{2})} + \frac{1}{3} \sec^{3} (\arctan (\frac{x}{2}))) + C$

Step 14.1.7

Pull terms out from under the radical.

$8 (- (\frac{1}{2} \sqrt{4 + x^{2}}) + \frac{1}{3} \sec^{3} (\arctan (\frac{x}{2}))) + C$

Step 14.1.8

Combine $\frac{1}{2}$ and $\sqrt{4 + x^{2}}$ .

$8 (- \frac{\sqrt{4 + x^{2}}}{2} + \frac{1}{3} \sec^{3} (\arctan (\frac{x}{2}))) + C$

Step 14.1.9

$8 (- \frac{\sqrt{4 + x^{2}}}{2} + \frac{1}{3} {\sqrt{1 + {(\frac{x}{2})}^{2}}}^{3}) + C$

Step 14.1.10

Apply the product rule to $\frac{x}{2}$ .

$8 (- \frac{\sqrt{4 + x^{2}}}{2} + \frac{1}{3} {\sqrt{1 + \frac{x^{2}}{2^{2}}}}^{3}) + C$

Step 14.1.11

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

$8 (- \frac{\sqrt{4 + x^{2}}}{2} + \frac{1}{3} {\sqrt{1 + \frac{x^{2}}{4}}}^{3}) + C$

Step 14.1.12

Write $1$ as a fraction with a common denominator.

$8 (- \frac{\sqrt{4 + x^{2}}}{2} + \frac{1}{3} {\sqrt{\frac{4}{4} + \frac{x^{2}}{4}}}^{3}) + C$

Step 14.1.13

Combine the numerators over the common denominator.

$8 (- \frac{\sqrt{4 + x^{2}}}{2} + \frac{1}{3} {\sqrt{\frac{4 + x^{2}}{4}}}^{3}) + C$

Step 14.1.14

Rewrite $\frac{4 + x^{2}}{4}$ as ${(\frac{1}{2})}^{2} (4 + x^{2})$ .

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Step 14.1.14.1

Factor the perfect power $1^{2}$ out of $4 + x^{2}$ .

$8 (- \frac{\sqrt{4 + x^{2}}}{2} + \frac{1}{3} {\sqrt{\frac{1^{2} (4 + x^{2})}{4}}}^{3}) + C$

Step 14.1.14.2

Factor the perfect power $2^{2}$ out of $4$ .

$8 (- \frac{\sqrt{4 + x^{2}}}{2} + \frac{1}{3} {\sqrt{\frac{1^{2} (4 + x^{2})}{2^{2} \cdot 1}}}^{3}) + C$

Step 14.1.14.3

Rearrange the fraction $\frac{1^{2} (4 + x^{2})}{2^{2} \cdot 1}$ .

$8 (- \frac{\sqrt{4 + x^{2}}}{2} + \frac{1}{3} {\sqrt{{(\frac{1}{2})}^{2} (4 + x^{2})}}^{3}) + C$

Step 14.1.15

Pull terms out from under the radical.

$8 (- \frac{\sqrt{4 + x^{2}}}{2} + \frac{1}{3} {(\frac{1}{2} \sqrt{4 + x^{2}})}^{3}) + C$

Step 14.1.16

Combine $\frac{1}{2}$ and $\sqrt{4 + x^{2}}$ .

$8 (- \frac{\sqrt{4 + x^{2}}}{2} + \frac{1}{3} {(\frac{\sqrt{4 + x^{2}}}{2})}^{3}) + C$

Step 14.1.17

Apply the product rule to $\frac{\sqrt{4 + x^{2}}}{2}$ .

$8 (- \frac{\sqrt{4 + x^{2}}}{2} + \frac{1}{3} \cdot \frac{{\sqrt{4 + x^{2}}}^{3}}{2^{3}}) + C$

Step 14.1.18

Combine.

$8 (- \frac{\sqrt{4 + x^{2}}}{2} + \frac{1 {\sqrt{4 + x^{2}}}^{3}}{3 \cdot 2^{3}}) + C$

Step 14.1.19

Multiply ${\sqrt{4 + x^{2}}}^{3}$ by $1$ .

$8 (- \frac{\sqrt{4 + x^{2}}}{2} + \frac{{\sqrt{4 + x^{2}}}^{3}}{3 \cdot 2^{3}}) + C$

Step 14.1.20

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

$8 (- \frac{\sqrt{4 + x^{2}}}{2} + \frac{{\sqrt{4 + x^{2}}}^{3}}{3 \cdot 8}) + C$

Step 14.1.21

Simplify the numerator.

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Step 14.1.21.1

Rewrite ${\sqrt{4 + x^{2}}}^{3}$ as $\sqrt{{(4 + x^{2})}^{3}}$ .

$8 (- \frac{\sqrt{4 + x^{2}}}{2} + \frac{\sqrt{{(4 + x^{2})}^{3}}}{3 \cdot 8}) + C$

Step 14.1.21.2

Factor out ${(4 + x^{2})}^{2}$ .

$8 (- \frac{\sqrt{4 + x^{2}}}{2} + \frac{\sqrt{{(4 + x^{2})}^{2} (4 + x^{2})}}{3 \cdot 8}) + C$

Step 14.1.21.3

Pull terms out from under the radical.

$8 (- \frac{\sqrt{4 + x^{2}}}{2} + \frac{(4 + x^{2}) \sqrt{4 + x^{2}}}{3 \cdot 8}) + C$

Step 14.1.21.4

Apply the distributive property.

$8 (- \frac{\sqrt{4 + x^{2}}}{2} + \frac{4 \sqrt{4 + x^{2}} + x^{2} \sqrt{4 + x^{2}}}{3 \cdot 8}) + C$

Step 14.1.21.5

Factor $\sqrt{4 + x^{2}}$ out of $4 \sqrt{4 + x^{2}} + x^{2} \sqrt{4 + x^{2}}$ .

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Step 14.1.21.5.1

Factor $\sqrt{4 + x^{2}}$ out of $4 \sqrt{4 + x^{2}}$ .

$8 (- \frac{\sqrt{4 + x^{2}}}{2} + \frac{\sqrt{4 + x^{2}} \cdot 4 + x^{2} \sqrt{4 + x^{2}}}{3 \cdot 8}) + C$

Step 14.1.21.5.2

Factor $\sqrt{4 + x^{2}}$ out of $x^{2} \sqrt{4 + x^{2}}$ .

$8 (- \frac{\sqrt{4 + x^{2}}}{2} + \frac{\sqrt{4 + x^{2}} \cdot 4 + \sqrt{4 + x^{2}} x^{2}}{3 \cdot 8}) + C$

Step 14.1.21.5.3

Factor $\sqrt{4 + x^{2}}$ out of $\sqrt{4 + x^{2}} \cdot 4 + \sqrt{4 + x^{2}} x^{2}$ .

$8 (- \frac{\sqrt{4 + x^{2}}}{2} + \frac{\sqrt{4 + x^{2}} (4 + x^{2})}{3 \cdot 8}) + C$

Step 14.1.22

Multiply $3$ by $8$ .

$8 (- \frac{\sqrt{4 + x^{2}}}{2} + \frac{\sqrt{4 + x^{2}} (4 + x^{2})}{24}) + C$

Step 14.2

To write $- \frac{\sqrt{4 + x^{2}}}{2}$ as a fraction with a common denominator, multiply by $\frac{12}{12}$ .

$8 (- \frac{\sqrt{4 + x^{2}}}{2} \cdot \frac{12}{12} + \frac{\sqrt{4 + x^{2}} (4 + x^{2})}{24}) + C$

Step 14.3

Write each expression with a common denominator of $24$ , by multiplying each by an appropriate factor of $1$ .

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Step 14.3.1

Multiply $\frac{\sqrt{4 + x^{2}}}{2}$ by $\frac{12}{12}$ .

$8 (- \frac{\sqrt{4 + x^{2}} \cdot 12}{2 \cdot 12} + \frac{\sqrt{4 + x^{2}} (4 + x^{2})}{24}) + C$

Step 14.3.2

Multiply $2$ by $12$ .

$8 (- \frac{\sqrt{4 + x^{2}} \cdot 12}{24} + \frac{\sqrt{4 + x^{2}} (4 + x^{2})}{24}) + C$

Step 14.4

Combine the numerators over the common denominator.

$8 \frac{- \sqrt{4 + x^{2}} \cdot 12 + \sqrt{4 + x^{2}} (4 + x^{2})}{24} + C$

Step 14.5

Cancel the common factor of $8$ .

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Step 14.5.1

Factor $8$ out of $24$ .

$8 \frac{- \sqrt{4 + x^{2}} \cdot 12 + \sqrt{4 + x^{2}} (4 + x^{2})}{8 (3)} + C$

Step 14.5.2

Cancel the common factor.

$8 \frac{- \sqrt{4 + x^{2}} \cdot 12 + \sqrt{4 + x^{2}} (4 + x^{2})}{8 \cdot 3} + C$

Step 14.5.3

Rewrite the expression.

$\frac{- \sqrt{4 + x^{2}} \cdot 12 + \sqrt{4 + x^{2}} (4 + x^{2})}{3} + C$

Step 14.6

Simplify the numerator.

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Step 14.6.1

Factor $\sqrt{4 + x^{2}}$ out of $- \sqrt{4 + x^{2}} \cdot 12 + \sqrt{4 + x^{2}} (4 + x^{2})$ .

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Step 14.6.1.1

Factor $\sqrt{4 + x^{2}}$ out of $- \sqrt{4 + x^{2}} \cdot 12$ .

$\frac{\sqrt{4 + x^{2}} (- 1 \cdot 12) + \sqrt{4 + x^{2}} (4 + x^{2})}{3} + C$

Step 14.6.1.2

Factor $\sqrt{4 + x^{2}}$ out of $\sqrt{4 + x^{2}} (- 1 \cdot 12) + \sqrt{4 + x^{2}} (4 + x^{2})$ .

$\frac{\sqrt{4 + x^{2}} (- 1 \cdot 12 + 4 + x^{2})}{3} + C$

Step 14.6.2

Multiply $- 1$ by $12$ .

$\frac{\sqrt{4 + x^{2}} (- 12 + 4 + x^{2})}{3} + C$

Step 14.6.3

Add $- 12$ and $4$ .

$\frac{\sqrt{4 + x^{2}} (- 8 + x^{2})}{3} + C$

Step 14.7

Rewrite $- 8$ as $- 1 (8)$ .

$\frac{\sqrt{4 + x^{2}} (- 1 (8) + x^{2})}{3} + C$

Step 14.8

Factor $- 1$ out of $x^{2}$ .

$\frac{\sqrt{4 + x^{2}} (- 1 (8) - 1 (- x^{2}))}{3} + C$

Step 14.9

Factor $- 1$ out of $- 1 (8) - 1 (- x^{2})$ .

$\frac{\sqrt{4 + x^{2}} (- 1 (8 - x^{2}))}{3} + C$

Step 14.10

Move the negative in front of the fraction.

$- \frac{\sqrt{4 + x^{2}} (8 - x^{2})}{3} + C$