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Chapter 5: Problem 43

Use symmetry to evaluate the following integrals. $$\int_{-\pi / 4}^{\pi / 4} \sec ^{2} x d x$$

Short Answer

Expert verified
Answer: The value of the definite integral is 2.

Step by step solution

01

Verify if the function is even or odd

First, we need to verify if the secant function is even or odd. An even function satisfies \(f(x) = f(-x)\), while an odd function satisfies \(f(x) = -f(-x)\). For the secant function, we have $$\sec (-x) = \frac{1}{\cos(-x)} = \frac{1}{\cos x} = \sec x.$$ Thus, the secant function is even. Since the integrand is \(\sec^2(x)\), it is also even.
02

Evaluate the integral of the even function

Since our integrand is even and our interval is symmetric with respect to the origin, we can use the property of even functions to rewrite our integral as: $$\int_{-\pi / 4}^{\pi / 4} \sec ^{2} x d x = 2\int_{0}^{\pi / 4} \sec ^{2} x d x.$$
03

Calculate the definite integral

Now, let's evaluate the definite integral: $$2\int_{0}^{\pi / 4} \sec ^{2} x d x.$$ The integral of \(\sec^2(x)\) is \(\tan(x)\), so we have: $$2(\tan(\pi/4) - \tan(0)).$$
04

Final Solution

Finally, let's calculate the value: $$2(\tan(\pi/4) - \tan(0)) = 2(1 - 0) = 2.$$ So, the value of the integral is \(2\).

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Most popular questions from this chapter

Substitutions Suppose that \(f\) is an even function with \(\int_{0}^{8} f(x) d x=9 .\) Evaluate each integral. a. \(\int_{-1}^{1} x f\left(x^{2}\right) d x\) b. \(\int_{-2}^{2} x^{2} f\left(x^{3}\right) d x\)

Integral of \(\sin ^{2} x \cos ^{2} x\) Consider the integral \(I=\int \sin ^{2} x \cos ^{2} x d x\) a. Find \(I\) using the identity \(\sin 2 x=2 \sin x \cos x\) b. Find \(I\) using the identity \(\cos ^{2} x=1-\sin ^{2} x\) c. Confirm that the results in parts (a) and (b) are consistent and compare the work involved in each method.

Find the following integrals. $$\int \frac{y^{2}}{(y+1)^{4}} d y$$

Use a change of variables to find the following indefinite integrals. Check your work by differentiating. $$\int\left(x^{6}-3 x^{2}\right)^{4}\left(x^{5}-x\right) d x$$

Suppose \(f\) is continuous on \([a, b]\) with \(f^{\prime \prime}(x)>0\) on the interval. It can be shown that $$(b-a) f\left(\frac{a+b}{2}\right) \leq \int_{a}^{b} f(x) d x \leq(b-a) \frac{f(a)+f(b)}{2}$$. a. Assuming \(f\) is nonnegative on \([a, b],\) draw a figure to illustrate the geometric meaning of these inequalities. Discuss your conclusions. b. Divide these inequalities by \((b-a)\) and interpret the resulting inequalities in terms of the average value of \(f\) on \([a, b]\).
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