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Chapter 4: Problem 21

Evaluate the integral. $$ \int_{-1 / 2}^{0} \frac{x}{\sqrt{1-x^{2}}} d x $$

Short Answer

Expert verified
The solution to the integral \(\int_{-1/2}^{0} \frac{x}{\sqrt{1-x^{2}}} dx\) is \(-1 + \sqrt{3} / 2\).

Step by step solution

01

Apply substitution method

Let \(u = 1 - x^2\). Therefore, the differential \(du = -2x\ dx\).
02

Rearrange the limits of integration

Substitute the original limits of \(x = -1/2\) and \(x = 0\) into \(u = 1 - x^2\) to get the new limits of integration for \(u\). The new limits are \(u(0) = 1\) and \(u(-1/2) = 3/4\).
03

Substitute \(u\) and \(du\) into the integral

The integral term becomes \(-1/2 \int_{3/4}^{1} u^{-1/2}\ du\).
04

Evaluate the integral

Now the integral can be solved. The integral of \(u^{-1/2}\) is \(2u^{1/2}\). Evaluating this between the limits \(1\) and \(3/4\) gives: -1/2[2(1^{1/2}) - 2(3/4)^{1/2}] = -1/2[2 - 2\sqrt{3}/2] = -1 + \sqrt{3} / 2.

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