Calculus Examples

$\int x^{3} \sqrt{x^{2} - 25} d x$

Step 1

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

$\int {(5 \sec (t))}^{3} \sqrt{{(5 \sec (t))}^{2} - 25} (5 \sec (t) \tan (t)) d t$

Step 2

Simplify terms.

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Step 2.1

Simplify $\sqrt{{(5 \sec (t))}^{2} - 25}$ .

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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 $5 \sec (t)$ .

$\int {(5 \sec (t))}^{3} \sqrt{5^{2} \sec^{2} (t) - 25} (5 \sec (t) \tan (t)) d t$

Step 2.1.1.2

Raise $5$ to the power of $2$ .

$\int {(5 \sec (t))}^{3} \sqrt{25 \sec^{2} (t) - 25} (5 \sec (t) \tan (t)) d t$

Step 2.1.2

Factor $25$ out of $25 \sec^{2} (t)$ .

$\int {(5 \sec (t))}^{3} \sqrt{25 (\sec^{2} (t)) - 25} (5 \sec (t) \tan (t)) d t$

Step 2.1.3

Factor $25$ out of $- 25$ .

$\int {(5 \sec (t))}^{3} \sqrt{25 \sec^{2} (t) + 25 \cdot - 1} (5 \sec (t) \tan (t)) d t$

Step 2.1.4

Factor $25$ out of $25 \sec^{2} (t) + 25 \cdot - 1$ .

$\int {(5 \sec (t))}^{3} \sqrt{25 (\sec^{2} (t) - 1)} (5 \sec (t) \tan (t)) d t$

Step 2.1.5

Apply pythagorean identity.

$\int {(5 \sec (t))}^{3} \sqrt{25 \tan^{2} (t)} (5 \sec (t) \tan (t)) d t$

Step 2.1.6

Rewrite $25 \tan^{2} (t)$ as ${(5 \tan (t))}^{2}$ .

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

Step 2.1.7

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

$\int {(5 \sec (t))}^{3} (5 \tan (t)) (5 \sec (t) \tan (t)) d t$

Step 2.2

Simplify.

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Step 2.2.1

Factor $5$ out of $5 \sec (t)$ .

$\int {(5 (\sec (t)))}^{3} (5 \tan (t)) (5 \sec (t) \tan (t)) d t$

Step 2.2.2

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

$\int 5^{3} \sec^{3} (t) (5 \tan (t)) (5 \sec (t) \tan (t)) d t$

Step 2.2.3

Raise $5$ to the power of $3$ .

$\int 125 \sec^{3} (t) (5 \tan (t)) (5 \sec (t) \tan (t)) d t$

Step 2.2.4

Multiply $5$ by $125$ .

$\int 625 \sec^{3} (t) \tan (t) (5 \sec (t) \tan (t)) d t$

Step 2.2.5

Multiply $5$ by $625$ .

$\int 3125 \sec^{3} (t) \tan (t) (\sec (t) \tan (t)) d t$

Step 2.2.6

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

$\int 3125 (\sec^{1} (t) \sec^{3} (t)) \tan (t) \tan (t) d t$

Step 2.2.7

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

$\int 3125 {\sec (t)}^{1 + 3} \tan (t) \tan (t) d t$

Step 2.2.8

Add $1$ and $3$ .

$\int 3125 \sec^{4} (t) \tan (t) \tan (t) d t$

Step 2.2.9

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

$\int 3125 \sec^{4} (t) (\tan^{1} (t) \tan (t)) d t$

Step 2.2.10

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

$\int 3125 \sec^{4} (t) (\tan^{1} (t) \tan^{1} (t)) d t$

Step 2.2.11

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

$\int 3125 \sec^{4} (t) {\tan (t)}^{1 + 1} d t$

Step 2.2.12

Add $1$ and $1$ .

$\int 3125 \sec^{4} (t) \tan^{2} (t) d t$

Step 3

Since $3125$ is constant with respect to $t$ , move $3125$ out of the integral.

$3125 \int \sec^{4} (t) \tan^{2} (t) d t$

Step 4

Simplify the expression.

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Step 4.1

Rewrite $4$ as $2$ plus $2$

$3125 \int {\sec (t)}^{2 + 2} \tan^{2} (t) d t$

Step 4.2

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

$3125 \int \sec^{2} (t) \sec^{2} (t) \tan^{2} (t) d t$

Step 5

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

$3125 \int (1 + \tan^{2} (t)) \sec^{2} (t) \tan^{2} (t) d t$

Step 6

Let $u = \tan (t)$ . Then $d u = \sec^{2} (t) d t$ , so $\frac{1}{\sec^{2} (t)} d u = d t$ . Rewrite using $u$ and $d$ $u$ .

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Step 6.1

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

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Step 6.1.1

Differentiate $\tan (t)$ .

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

Step 6.1.2

The derivative of $\tan (t)$ with respect to $t$ is $\sec^{2} (t)$ .

$\sec^{2} (t)$

Step 6.2

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

$3125 \int (1 + u^{2}) u^{2} d u$

Step 7

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

$3125 \int 1 u^{2} + u^{2} u^{2} d u$

Step 8

Simplify.

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Step 8.1

Multiply $u^{2}$ by $1$ .

$3125 \int u^{2} + u^{2} u^{2} d u$

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.

$3125 \int u^{2} + u^{2 + 2} d u$

Step 8.2.2

Add $2$ and $2$ .

$3125 \int u^{2} + u^{4} d u$

Step 9

Split the single integral into multiple integrals.

$3125 (\int u^{2} d u + \int u^{4} d u)$

Step 10

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

$3125 (\frac{1}{3} u^{3} + C + \int u^{4} d u)$

Step 11

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

$3125 (\frac{1}{3} u^{3} + C + \frac{1}{5} u^{5} + C)$

Step 12

Simplify.

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Step 12.1

Simplify.

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Step 12.1.1

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

$3125 (\frac{u^{3}}{3} + C + \frac{1}{5} u^{5} + C)$

Step 12.1.2

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

$3125 (\frac{u^{3}}{3} + C + \frac{u^{5}}{5} + C)$

Step 12.2

Simplify.

$3125 (\frac{1}{3} u^{3} + \frac{1}{5} u^{5}) + C$

Step 13

Substitute back in for each integration substitution variable.

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Step 13.1

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

$3125 (\frac{1}{3} \tan^{3} (t) + \frac{1}{5} \tan^{5} (t)) + C$

Step 13.2

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

$3125 (\frac{1}{3} \tan^{3} (arcsec (\frac{x}{5})) + \frac{1}{5} \tan^{5} (arcsec (\frac{x}{5}))) + C$

Step 14

Reorder terms.

$3125 (\frac{1}{3} \tan^{3} (arcsec (\frac{1}{5} x)) + \frac{1}{5} \tan^{5} (arcsec (\frac{1}{5} x))) + C$