Problem 21

Question

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

Step-by-Step Solution

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Answer

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

1Step 1: Apply substitution method

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

2Step 2: 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 $u(-1/2) = 3/4$.

3Step 3: Substitute $u$ and $du$ into the integral

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

4Step 4: 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.

Problem 21

Other exercises in this chapter

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Problem 21

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