Calculus Examples

$\int \frac{1}{{(1 - x^{2})}^{\frac{3}{2}}} d x$

Step 1

Apply the rule $x^{\frac{m}{n}} = \sqrt[n]{x^{m}}$ to rewrite the exponentiation as a radical.

$\int \frac{1}{\sqrt{{(1 - x^{2})}^{3}}} d x$

Step 2

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.

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

Step 3

Simplify terms.

Tap for more steps...

Step 3.1

Simplify $\sqrt{{(1 - \sin^{2} (t))}^{3}}$ .

Tap for more steps...

Step 3.1.1

Apply pythagorean identity.

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

Step 3.1.2

Multiply the exponents in ${(\cos^{2} (t))}^{3}$ .

Tap for more steps...

Step 3.1.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 3.1.2.2

Multiply $2$ by $3$ .

$\int \frac{1}{\sqrt{\cos^{6} (t)}} \cos (t) d t$

Step 3.1.3

Rewrite $\cos^{6} (t)$ as ${(\cos^{3} (t))}^{2}$ .

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

Step 3.1.4

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

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

Step 3.2

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

Tap for more steps...

Step 3.2.1

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

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

Step 3.2.2

Cancel the common factor.

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

Step 3.2.3

Rewrite the expression.

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

Step 4

Convert from $\frac{1}{\cos^{2} (t)}$ to $\sec^{2} (t)$ .

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

Step 5

Since the derivative of $\tan (t)$ is $\sec^{2} (t)$ , the integral of $\sec^{2} (t)$ is $\tan (t)$ .

$\tan (t) + C$

Step 6

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

$\tan (\arcsin (x)) + C$