Calculus Examples

$\int \frac{\sqrt{x^{2} - 9}}{x^{3}} d x$

Step 1

Let $x = 3 \sec (t)$ , where $- \frac{π}{2} \leq t \leq \frac{π}{2}$ . Then $d x = 3 \sec (t) \tan (t) d t$ . Note that since $- \frac{π}{2} \leq t \leq \frac{π}{2}$ , $3 \sec (t) \tan (t)$ is positive.

$\int \frac{\sqrt{{(3 \sec (t))}^{2} - 9}}{{(3 \sec (t))}^{3}} (3 \sec (t) \tan (t)) d t$

Step 2

Simplify terms.

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Step 2.1

Simplify $\sqrt{{(3 \sec (t))}^{2} - 9}$ .

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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 $3 \sec (t)$ .

$\int \frac{\sqrt{3^{2} \sec^{2} (t) - 9}}{{(3 \sec (t))}^{3}} (3 \sec (t) \tan (t)) d t$

Step 2.1.1.2

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

$\int \frac{\sqrt{9 \sec^{2} (t) - 9}}{{(3 \sec (t))}^{3}} (3 \sec (t) \tan (t)) d t$

Step 2.1.2

Factor $9$ out of $9 \sec^{2} (t)$ .

$\int \frac{\sqrt{9 (\sec^{2} (t)) - 9}}{{(3 \sec (t))}^{3}} (3 \sec (t) \tan (t)) d t$

Step 2.1.3

Factor $9$ out of $- 9$ .

$\int \frac{\sqrt{9 \sec^{2} (t) + 9 \cdot - 1}}{{(3 \sec (t))}^{3}} (3 \sec (t) \tan (t)) d t$

Step 2.1.4

Factor $9$ out of $9 \sec^{2} (t) + 9 \cdot - 1$ .

$\int \frac{\sqrt{9 (\sec^{2} (t) - 1)}}{{(3 \sec (t))}^{3}} (3 \sec (t) \tan (t)) d t$

Step 2.1.5

Apply pythagorean identity.

$\int \frac{\sqrt{9 \tan^{2} (t)}}{{(3 \sec (t))}^{3}} (3 \sec (t) \tan (t)) d t$

Step 2.1.6

Rewrite $9 \tan^{2} (t)$ as ${(3 \tan (t))}^{2}$ .

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

Step 2.1.7

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

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

Step 2.2

Simplify.

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Step 2.2.1

Factor $3$ out of $3 \sec (t)$ .

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

Step 2.2.2

Apply the product rule to $3 (\sec (t))$ .

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

Step 2.2.3

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

$\int \frac{3 \tan (t)}{27 \sec^{3} (t)} (3 \sec (t) \tan (t)) d t$

Step 2.2.4

Cancel the common factor of $3$ and $27$ .

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Step 2.2.4.1

Factor $3$ out of $3 \tan (t)$ .

$\int \frac{3 (\tan (t))}{27 \sec^{3} (t)} (3 \sec (t) \tan (t)) d t$

Step 2.2.4.2

Cancel the common factors.

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Step 2.2.4.2.1

Factor $3$ out of $27 \sec^{3} (t)$ .

$\int \frac{3 (\tan (t))}{3 (9 \sec^{3} (t))} (3 \sec (t) \tan (t)) d t$

Step 2.2.4.2.2

Cancel the common factor.

$\int \frac{3 \tan (t)}{3 (9 \sec^{3} (t))} (3 \sec (t) \tan (t)) d t$

Step 2.2.4.2.3

Rewrite the expression.

$\int \frac{\tan (t)}{9 \sec^{3} (t)} (3 \sec (t) \tan (t)) d t$

Step 2.2.5

Combine $3$ and $\frac{\tan (t)}{9 \sec^{3} (t)}$ .

$\int \frac{3 \tan (t)}{9 \sec^{3} (t)} (\sec (t) \tan (t)) d t$

Step 2.2.6

Combine $\sec (t)$ and $\frac{3 \tan (t)}{9 \sec^{3} (t)}$ .

$\int \frac{\sec (t) (3 \tan (t))}{9 \sec^{3} (t)} \tan (t) d t$

Step 2.2.7

Combine $\frac{\sec (t) (3 \tan (t))}{9 \sec^{3} (t)}$ and $\tan (t)$ .

$\int \frac{\sec (t) (3 \tan (t)) \tan (t)}{9 \sec^{3} (t)} d t$

Step 2.2.8

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

$\int \frac{\sec (t) (3 (\tan^{1} (t) \tan (t)))}{9 \sec^{3} (t)} d t$

Step 2.2.9

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

$\int \frac{\sec (t) (3 (\tan^{1} (t) \tan^{1} (t)))}{9 \sec^{3} (t)} d t$

Step 2.2.10

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

$\int \frac{\sec (t) (3 {\tan (t)}^{1 + 1})}{9 \sec^{3} (t)} d t$

Step 2.2.11

Add $1$ and $1$ .

$\int \frac{\sec (t) (3 \tan^{2} (t))}{9 \sec^{3} (t)} d t$

Step 2.2.12

Move $3$ to the left of $\sec (t)$ .

$\int \frac{3 \cdot \sec (t) \tan^{2} (t)}{9 \sec^{3} (t)} d t$

Step 2.2.13

Cancel the common factor of $3$ and $9$ .

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Step 2.2.13.1

Factor $3$ out of $3 \sec (t) \tan^{2} (t)$ .

$\int \frac{3 (\sec (t) \tan^{2} (t))}{9 \sec^{3} (t)} d t$

Step 2.2.13.2

Cancel the common factors.

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Step 2.2.13.2.1

Factor $3$ out of $9 \sec^{3} (t)$ .

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

Step 2.2.13.2.2

Cancel the common factor.

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

Step 2.2.13.2.3

Rewrite the expression.

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

Step 2.2.14

Cancel the common factor of $\sec (t)$ and $\sec^{3} (t)$ .

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Step 2.2.14.1

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

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

Step 2.2.14.2

Cancel the common factors.

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Step 2.2.14.2.1

Factor $\sec (t)$ out of $3 \sec^{3} (t)$ .

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

Step 2.2.14.2.2

Cancel the common factor.

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

Step 2.2.14.2.3

Rewrite the expression.

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

Step 3

Since $\frac{1}{3}$ is constant with respect to $t$ , move $\frac{1}{3}$ out of the integral.

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

Step 4

Simplify.

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Step 4.1

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

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

Step 4.2

Rewrite $\sec (t)$ in terms of sines and cosines.

$\frac{1}{3} \int {(\frac{\tan (t)}{\frac{1}{\cos (t)}})}^{2} d t$

Step 4.3

Rewrite $\tan (t)$ in terms of sines and cosines.

$\frac{1}{3} \int {(\frac{\frac{\sin (t)}{\cos (t)}}{\frac{1}{\cos (t)}})}^{2} d t$

Step 4.4

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\cos (t)}$ .

$\frac{1}{3} \int {(\frac{\sin (t)}{\cos (t)} \cos (t))}^{2} d t$

Step 4.5

Write $\cos (t)$ as a fraction with denominator $1$ .

$\frac{1}{3} \int {(\frac{\sin (t)}{\cos (t)} \cdot \frac{\cos (t)}{1})}^{2} d t$

Step 4.6

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

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Step 4.6.1

Cancel the common factor.

$\frac{1}{3} \int {(\frac{\sin (t)}{\cos (t)} \cdot \frac{\cos (t)}{1})}^{2} d t$

Step 4.6.2

Rewrite the expression.

$\frac{1}{3} \int \sin^{2} (t) d t$

Step 5

Use the half-angle formula to rewrite $\sin^{2} (t)$ as $\frac{1 - \cos (2 t)}{2}$ .

$\frac{1}{3} \int \frac{1 - \cos (2 t)}{2} d t$

Step 6

Since $\frac{1}{2}$ is constant with respect to $t$ , move $\frac{1}{2}$ out of the integral.

$\frac{1}{3} (\frac{1}{2} \int 1 - \cos (2 t) d t)$

Step 7

Simplify.

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Step 7.1

Multiply $\frac{1}{2}$ by $\frac{1}{3}$ .

$\frac{1}{2 \cdot 3} \int 1 - \cos (2 t) d t$

Step 7.2

Multiply $2$ by $3$ .

$\frac{1}{6} \int 1 - \cos (2 t) d t$

Step 8

Split the single integral into multiple integrals.

$\frac{1}{6} (\int d t + \int - \cos (2 t) d t)$

Step 9

Apply the constant rule.

$\frac{1}{6} (t + C + \int - \cos (2 t) d t)$

Step 10

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

$\frac{1}{6} (t + C - \int \cos (2 t) d t)$

Step 11

Let $u = 2 t$ . Then $d u = 2 d t$ , so $\frac{1}{2} d u = d t$ . Rewrite using $u$ and $d$ $u$ .

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Step 11.1

Let $u = 2 t$ . Find $\frac{d u}{d t}$ .

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Step 11.1.1

Differentiate $2 t$ .

$\frac{d}{d t} [2 t]$

Step 11.1.2

Since $2$ is constant with respect to $t$ , the derivative of $2 t$ with respect to $t$ is $2 \frac{d}{d t} [t]$ .

$2 \frac{d}{d t} [t]$

Step 11.1.3

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 1$ .

$2 \cdot 1$

Step 11.1.4

Multiply $2$ by $1$ .

$2$

Step 11.2

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

$\frac{1}{6} (t + C - \int \cos (u) \frac{1}{2} d u)$

Step 12

Combine $\cos (u)$ and $\frac{1}{2}$ .

$\frac{1}{6} (t + C - \int \frac{\cos (u)}{2} d u)$

Step 13

Since $\frac{1}{2}$ is constant with respect to $u$ , move $\frac{1}{2}$ out of the integral.

$\frac{1}{6} (t + C - (\frac{1}{2} \int \cos (u) d u))$

Step 14

The integral of $\cos (u)$ with respect to $u$ is $\sin (u)$ .

$\frac{1}{6} (t + C - \frac{1}{2} (\sin (u) + C))$

Step 15

Simplify.

$\frac{1}{6} (t - \frac{1}{2} \sin (u)) + C$

Step 16

Substitute back in for each integration substitution variable.

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Step 16.1

Replace all occurrences of $t$ with $arcsec (\frac{x}{3})$ .

$\frac{1}{6} (arcsec (\frac{x}{3}) - \frac{1}{2} \sin (u)) + C$

Step 16.2

Replace all occurrences of $u$ with $2 t$ .

$\frac{1}{6} (arcsec (\frac{x}{3}) - \frac{1}{2} \sin (2 t)) + C$

Step 16.3

Replace all occurrences of $t$ with $arcsec (\frac{x}{3})$ .

$\frac{1}{6} (arcsec (\frac{x}{3}) - \frac{1}{2} \sin (2 arcsec (\frac{x}{3}))) + C$

Step 17

Simplify.

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Step 17.1

Combine $\sin (2 arcsec (\frac{x}{3}))$ and $\frac{1}{2}$ .

$\frac{1}{6} (arcsec (\frac{x}{3}) - \frac{\sin (2 arcsec (\frac{x}{3}))}{2}) + C$

Step 17.2

Apply the distributive property.

$\frac{1}{6} arcsec (\frac{x}{3}) + \frac{1}{6} (- \frac{\sin (2 arcsec (\frac{x}{3}))}{2}) + C$

Step 17.3

Combine $\frac{1}{6}$ and $arcsec (\frac{x}{3})$ .

$\frac{arcsec (\frac{x}{3})}{6} + \frac{1}{6} (- \frac{\sin (2 arcsec (\frac{x}{3}))}{2}) + C$

Step 17.4

Multiply $\frac{1}{6} (- \frac{\sin (2 arcsec (\frac{x}{3}))}{2})$ .

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Step 17.4.1

Multiply $\frac{1}{6}$ by $\frac{\sin (2 arcsec (\frac{x}{3}))}{2}$ .

$\frac{arcsec (\frac{x}{3})}{6} - \frac{\sin (2 arcsec (\frac{x}{3}))}{6 \cdot 2} + C$

Step 17.4.2

Multiply $6$ by $2$ .

$\frac{arcsec (\frac{x}{3})}{6} - \frac{\sin (2 arcsec (\frac{x}{3}))}{12} + C$

Step 18

Reorder terms.

$\frac{1}{6} arcsec (\frac{1}{3} x) - \frac{1}{12} \sin (2 arcsec (\frac{1}{3} x)) + C$