Calculus Examples

$\int \frac{1}{\sqrt{9 - x^{2}}} d x$

Step 1

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.

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

Step 2

Simplify terms.

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Step 2.1

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

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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 $3 \sin (t)$ .

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

Step 2.1.1.2

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

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

Step 2.1.1.3

Multiply $9$ by $- 1$ .

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

Step 2.1.2

Factor $9$ out of $9$ .

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

Step 2.1.3

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

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

Step 2.1.4

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

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

Step 2.1.5

Apply pythagorean identity.

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

Step 2.1.6

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

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

Step 2.1.7

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

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

Step 2.2

Cancel the common factor of $3 \cos (t)$ .

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Step 2.2.1

Cancel the common factor.

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

Step 2.2.2

Rewrite the expression.

$\int d t$

Step 3

Apply the constant rule.

$t + C$

Step 4

Simplify the answer.

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Step 4.1

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

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

Step 4.2

Reorder terms.

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