Calculus Examples

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

Step 1

This integral could not be completed using u-substitution. Mathway will use another method.

Step 2

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

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

Step 3

Simplify terms.

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Step 3.1

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

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Step 3.1.1

Simplify each term.

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Step 3.1.1.1

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

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

Step 3.1.1.2

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

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

Step 3.1.1.3

Multiply $9$ by $- 1$ .

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

Step 3.1.2

Factor $9$ out of $9$ .

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

Step 3.1.3

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

$\int \frac{\sqrt{9 (1) + 9 (- \sin^{2} (t))}}{{(3 \sin (t))}^{2}} (3 \cos (t)) d t$

Step 3.1.4

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

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

Step 3.1.5

Apply pythagorean identity.

$\int \frac{\sqrt{9 \cos^{2} (t)}}{{(3 \sin (t))}^{2}} (3 \cos (t)) d t$

Step 3.1.6

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

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

Step 3.1.7

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

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

Step 3.2

Simplify.

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.2.3

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

$\int \frac{3 \cos (t)}{9 \sin^{2} (t)} (3 \cos (t)) d t$

Step 3.2.4

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

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Step 3.2.4.1

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

$\int \frac{3 (\cos (t))}{9 \sin^{2} (t)} (3 \cos (t)) d t$

Step 3.2.4.2

Cancel the common factors.

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Step 3.2.4.2.1

Factor $3$ out of $9 \sin^{2} (t)$ .

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

Step 3.2.4.2.2

Cancel the common factor.

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

Step 3.2.4.2.3

Rewrite the expression.

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

Step 3.2.5

Combine $3$ and $\frac{\cos (t)}{3 \sin^{2} (t)}$ .

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

Step 3.2.6

Combine $\frac{3 \cos (t)}{3 \sin^{2} (t)}$ and $\cos (t)$ .

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

Step 3.2.7

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

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

Step 3.2.8

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

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

Step 3.2.9

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

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

Step 3.2.10

Add $1$ and $1$ .

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

Step 3.2.11

Cancel the common factor of $3$ .

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Step 3.2.11.1

Cancel the common factor.

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

Step 3.2.11.2

Rewrite the expression.

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

Step 3.2.12

Convert from $\frac{\cos^{2} (t)}{\sin^{2} (t)}$ to $\cot^{2} (t)$ .

$\int \cot^{2} (t) d t$

Step 4

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

$\int - 1 + \csc^{2} (t) d t$

Step 5

Split the single integral into multiple integrals.

$\int - 1 d t + \int \csc^{2} (t) d t$

Step 6

Apply the constant rule.

$- t + C + \int \csc^{2} (t) d t$

Step 7

Since the derivative of $- \cot (t)$ is $\csc^{2} (t)$ , the integral of $\csc^{2} (t)$ is $- \cot (t)$ .

$- t + C - \cot (t) + C$

Step 8

Simplify.

$- t - \cot (t) + C$

Step 9

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

$- \arcsin (\frac{x}{3}) - \cot (\arcsin (\frac{x}{3})) + C$

Step 10

Reorder terms.

$- \arcsin (\frac{1}{3} x) - \cot (\arcsin (\frac{1}{3} x)) + C$