Calculus Examples

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

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.

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

Step 2

Simplify terms.

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Step 2.1

Simplify $\sqrt{1 - \sin^{2} (t)}$ .

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Step 2.1.1

Apply pythagorean identity.

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

Step 2.1.2

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

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

Step 2.2

Simplify.

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

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

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

Step 2.2.4

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

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

Step 2.2.5

Add $1$ and $1$ .

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

Step 2.2.6

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

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

Step 3

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

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

Step 4

Split the single integral into multiple integrals.

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

Step 5

Apply the constant rule.

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

Step 6

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 7

Simplify.

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

Step 8

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

$- \arcsin (x) - \cot (\arcsin (x)) + C$