Calculus Examples

$\int \frac{1}{\sqrt{4 - 4 x^{2}}} d x$

Step 1

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

$\int \frac{1}{\sqrt{4 - 4 \sin^{2} (t)}} \cos (t) d t$

Step 2

Simplify terms.

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Step 2.1

Simplify $\sqrt{4 - 4 \sin^{2} (t)}$ .

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Step 2.1.1

Factor $4$ out of $4$ .

$\int \frac{1}{\sqrt{4 (1) - 4 \sin^{2} (t)}} \cos (t) d t$

Step 2.1.2

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

$\int \frac{1}{\sqrt{4 (1) + 4 (- \sin^{2} (t))}} \cos (t) d t$

Step 2.1.3

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

$\int \frac{1}{\sqrt{4 (1 - \sin^{2} (t))}} \cos (t) d t$

Step 2.1.4

Apply pythagorean identity.

$\int \frac{1}{\sqrt{4 \cos^{2} (t)}} \cos (t) d t$

Step 2.1.5

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

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

Step 2.1.6

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

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

Cancel the common factor.

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

Step 2.2.3

Rewrite the expression.

$\int \frac{1}{2} d t$

Step 3

Apply the constant rule.

$\frac{1}{2} t + C$

Step 4

Replace all occurrences of $t$ with $\arcsin (x)$ .

$\frac{1}{2} \arcsin (x) + C$