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Chapter 7: Problem 37

Evaluate the following definite integrals. $$\int_{1 / 2}^{\sqrt{3} / 2} \sin ^{-1} y d y$$

Short Answer

Expert verified
Question: Evaluate the definite integral of the inverse sine function with respect to y from the lower limit of integration 1/2 to the upper limit of integration sqrt(3)/2. Answer: The definite integral of the inverse sine function with respect to y from 1/2 to sqrt(3)/2 is equal to $$\frac{1}{2}\left[ \frac{\pi}{3} + \frac{\pi}{4} - 1 - \sqrt{2} + \frac{1}{3} \right]$$.

Step by step solution

01

Integration by parts

To evaluate the given integral, we need to use integration by parts. Integration by parts is a technique used to integrate a product of two functions. In this case, we'll let: u = arcsin(y) and dv = dy Now, we'll find the derivatives and antiderivatives: du = 1/鈭�(1-y^2) dy v = y Now we can apply the formula for integration by parts: 鈭玼 dv = uv - 鈭玽 du
02

Applying the formula

Applying the integration by parts formula, we obtain: $$\int_{1/2}^{\sqrt{3}/2} sin^{-1} (y) dy = \left[ y \sin ^{-1} y \right]_{1/2}^{\sqrt{3}/2} - \int_{1/2}^{\sqrt{3}/2}\frac{y}{\sqrt{1 - y^{2}}} dy$$
03

Use Substitution

To solve the second integral, we will use the substitution method. Let: x = y^2 鈫� dx = 2y dy y = 鈭歺 Now substitute these expressions and adjust the limits of integration: $$- \int_{1/4}^{3/4} \frac{\sqrt{x}}{\sqrt{1-x}} 2 \sqrt{x} dx$$ We notice that the 2 in the numerator and denominator gets canceled out.
04

Simplifying and integrating

Now we can simplify the integral: $$- \int_{1/4}^{3/4} \frac{x}{\sqrt{1-x}} dx$$ We can integrate this using the power rule for integration: $$- \left[ x\sqrt{1-x} + \frac{1}{3}(1-x)^{3/2} \right]_{1/4}^{3/4}$$
05

Evaluate the definite integral

Now, we can evaluate the definite integral by plugging in the limits of integration: $$\left[ \frac{\sqrt{3}}{2} \sin ^{-1} \frac{\sqrt{3}}{2} - \frac{1}{2} \sin ^{-1} \frac{1}{2} \right] - \left[ \left( \frac{3}{4} \right)\left( \frac{1}{2} \right) + \frac{1}{3}\left(\frac{1}{4}\right)^{3/2}- \left(\frac{1}{4}\right)\left(\frac{1}{\sqrt{2}}\right) - \frac{1}{3}\left(\frac{1}{2}\right)^{3/2} \right]$$ After evaluating this expression, we will get the final answer to the definite integral: $$\frac{1}{2}\left[ \frac{\pi}{3} + \frac{\pi}{4} - 1 - \sqrt{2} + \frac{1}{3} \right]$$

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Most popular questions from this chapter

Evaluate the following integrals. Consider completing the square. $$\int_{2+\sqrt{2}}^{4} \frac{d x}{\sqrt{(x-1)(x-3)}}$$

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