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Chapter 4: Problem 22

Evaluate the integral. $$ \int_{0}^{\pi / 2} \frac{\cos x}{1+\sin ^{2} x} d x $$

Short Answer

Expert verified
The value of the integral is \(\pi / 4\).

Step by step solution

01

Setting Up the Substitution

In attempt to simplify the integral, let's make a substitution. We substitute \(u = \sin x\) which implies \(du = \cos x \, dx\). Replacing \(\cos x \, dx\) with \(du\) makes the integral simpler. But we need to make sure to change the limits of integration as well. The initial lower limit was \(x = 0\) which now translates to \(u = \sin 0 = 0\), and the initial upper limit was \(x = \pi / 2\) which now translates to \(u = \sin (\pi / 2) = 1\).
02

Substitute Into the Integral

Now our integral, with the substitution and the new limits of integration, becomes: \(\int_{0}^{1} \frac{1}{1+u^2} du\). This is a simpler integral to calculate.
03

Evaluating the Integral

The integral \(\int \frac{1}{1+u^2} du\) is a standard integral whose result is \(\arctan(u)\). Applying this rule, we get \(\arctan(u)\) from 0 to 1.
04

Evaluating the Limits

The final step involves substituting the limits of integration back into the result, and subtracting to find the final answer: \(\arctan(1) - \arctan(0) = \pi/4 - 0 = \pi/4\).

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Most popular questions from this chapter

A horizontal plane is ruled with parallel lines 2 inches apart. A two-inch needle is tossed randomly onto the plane. The probability that the needle will touch a line is \(P=\frac{2}{\pi} \int_{0}^{\pi / 2} \sin \theta d \theta\) where \(\theta\) is the acute angle between the needle and any one of the parallel lines. Find this probability.

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