Problem 34 - Essential Calculus

Question

Evaluate the integral using (a) the given integration limits and (b) the limits obtained by trigonometric substitution. $$ \int_{0}^{\sqrt{3} / 2} \frac{1}{\left(1-t^{2}\right)^{5 / 2}} d t $$

Step-by-Step Solution

Answer

After evaluation, the integral is equal to \( \frac{3}{8}\theta + \frac{1}{4}sin(2\theta) + \frac{1}{32}sin(4\theta) \) evaluated from 0 to \( \pi / 3 \) and comes out to be \( \frac{1}{32} \pi - \frac{1}{16} \).

Step 1: Identify Substitution

In the equation, we will start by identifying a suitable substitution. We can set \( t = sin\theta \).

Step 2: Change the Limits

Substitute the old limits (0 to \( \sqrt{3} / 2 \)) with the new limits. These are obtained by substituting the old limits into the substitution identified in step 1. So we have \( 0 -> sin^{-1} 0 = 0 \) and \( \sqrt{3} / 2 -> sin^{-1} \sqrt{3}/2 = \pi / 3 \). Therefore, new limits of integration are 0 to \( \pi / 3 \).

Step 3: Substitution in Integral

Now, substitute \( t = sin\theta \) into the original integral and transform \( dt \) to \( cos\theta d\theta \). It becomes \( \int \frac{cos\theta d\theta}{(1 - sin^{2}\theta)^{5 / 2}} \). Using Pythagorean identity it simplifies to \( \int cos^{4}\theta d\theta \)

Step 4: Evaluate the Integral

Using the power-reducing identity, evaluate the integral. It becomes \( \int \frac{3}{8} + \frac{1}{2}cos(2\theta) + \frac{1}{8}cos(4\theta) d\theta \). The function can be integrated from 0 to \( \pi / 3 \) resulting in the final answer.