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Chapter 5: Problem 39

Evaluate the following integrals using the Fundamental Theorem of Calculus. $$\int_{0}^{\pi / 4} \sec ^{2} \theta d \theta$$

Short Answer

Expert verified
Question: Evaluate the definite integral $$\int_{0}^{\pi / 4} \sec ^{2} \theta d \theta$$. Answer: The definite integral $$\int_{0}^{\pi / 4} \sec ^{2} \theta d \theta$$ is equal to 1.

Step by step solution

01

Find the antiderivative of $$\sec^{2}\theta$$

To find the antiderivative of $$\sec^{2}\theta$$, recall that the derivative of $$\tan\theta$$ is $$\sec^{2}\theta$$. Therefore, the antiderivative of $$\sec^2\theta$$ is $$\tan\theta$$.
02

Evaluate \(\tan\theta\) at the limits

We need to find the values of $$\tan\theta$$ at $$\theta = \frac{\pi}{4}$$ and $$\theta = 0$$. When $$\theta = \frac{\pi}{4}$$, we have $$\tan\frac{\pi}{4} = 1$$. When $$\theta = 0$$, we have $$\tan 0 = 0$$.
03

Apply the Fundamental Theorem of Calculus

Using the Fundamental Theorem of Calculus, we can evaluate the definite integral by subtracting the value of the antiderivative at the lower limit from the upper limit: $$\int_{0}^{\pi / 4} \sec ^{2} \theta d \theta = \left[\tan \theta\right]_0^{\frac{\pi}{4}} = \tan\frac{\pi}{4} - \tan(0) = 1 - 0 = 1$$ Therefore, the value of the definite integral $$\int_{0}^{\pi / 4} \sec ^{2} \theta d \theta$$ is 1.

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