Calculus Examples

$\int_{\frac{1}{2}}^{\sqrt{\frac{3}{2}}} \frac{1}{x^{2} \sqrt{1 - x^{2}}} d x$

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.

$\int_{\frac{π}{6}}^{\arcsin (\frac{\sqrt{6}}{2})} \frac{1}{\sin^{2} (t) \sqrt{1 - \sin^{2} (t)}} \cos (t) d t$

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.

$\int_{\frac{π}{6}}^{\arcsin (\frac{\sqrt{6}}{2})} \frac{1}{\sin^{2} (t) \sqrt{\cos^{2} (t)}} \cos (t) d t$

Step 2.1.2

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

$\int_{\frac{π}{6}}^{\arcsin (\frac{\sqrt{6}}{2})} \frac{1}{\sin^{2} (t) \cos (t)} \cos (t) d t$

Step 2.2

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

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Step 2.2.1

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

$\int_{\frac{π}{6}}^{\arcsin (\frac{\sqrt{6}}{2})} \frac{1}{\cos (t) \sin^{2} (t)} \cos (t) d t$

Step 2.2.2

Cancel the common factor.

$\int_{\frac{π}{6}}^{\arcsin (\frac{\sqrt{6}}{2})} \frac{1}{\cos (t) \sin^{2} (t)} \cos (t) d t$

Step 2.2.3

Rewrite the expression.

$\int_{\frac{π}{6}}^{\arcsin (\frac{\sqrt{6}}{2})} \frac{1}{\sin^{2} (t)} d t$

Step 3

Convert from $\frac{1}{\sin^{2} (t)}$ to $\csc^{2} (t)$ .

$\int_{\frac{π}{6}}^{\arcsin (\frac{\sqrt{6}}{2})} \csc^{2} (t) d t$

Step 4

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

$- \cot (t)]_{\frac{π}{6}}^{\arcsin (\frac{\sqrt{6}}{2})}$

Step 5

Substitute and simplify.

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Step 5.1

Evaluate $- \cot (t)$ at $\arcsin (\frac{\sqrt{6}}{2})$ and at $\frac{π}{6}$ .

$(- \cot (\arcsin (\frac{\sqrt{6}}{2}))) + \cot (\frac{π}{6})$

Step 5.2

Draw a triangle in the plane with vertices $(\sqrt{1^{2} - {(\frac{\sqrt{6}}{2})}^{2}}, \frac{\sqrt{6}}{2})$ , $(\sqrt{1^{2} - {(\frac{\sqrt{6}}{2})}^{2}}, 0)$ , and the origin. Then $\arcsin (\frac{\sqrt{6}}{2})$ is the angle between the positive x-axis and the ray beginning at the origin and passing through $(\sqrt{1^{2} - {(\frac{\sqrt{6}}{2})}^{2}}, \frac{\sqrt{6}}{2})$ . Therefore, $\cot (\arcsin (\frac{\sqrt{6}}{2}))$ is $\frac{i \sqrt{3}}{3}$ .

$- \frac{i \sqrt{3}}{3} + \cot (\frac{π}{6})$

Step 6

The exact value of $\cot (\frac{π}{6})$ is $\sqrt{3}$ .

$- \frac{i \sqrt{3}}{3} + \sqrt{3}$

Step 7