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Chapter 5: Problem 90

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

Short Answer

Expert verified
Question: Find the value of the definite integral: $$\int_{\pi / 4}^{\pi / 2} \csc^2 \theta \, d\theta$$ Answer: The value of the definite integral is 1.

Step by step solution

01

Find the antiderivative of \(\csc^2\theta\)

Recall that the antiderivative of \(\csc^2\theta\) is \(-\cot\theta + C\), where \(C\) is the constant of integration.
02

Apply the Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus states that if `F` is the antiderivative of a continuous function `f` over the interval `[a, b]`, then: $$\int_{a}^{b} f(x) \, dx = F(b) - F(a)$$ In our case, \(f(\theta) = \csc^2\theta\) and \(F(\theta) = -\cot\theta\). We have the limits \(a = \frac{\pi}{4}\) and \(b = \frac{\pi}{2}\). Therefore, we can write the definite integral as: $$\int_{\pi / 4}^{\pi / 2} \csc^2 \theta \, d\theta = -\cot\left(\frac{\pi}{2}\right) - (-\cot\left(\frac{\pi}{4}\right))$$
03

Evaluate the expression and simplify

Now, we plug in the values of the limits and evaluate the expression: $$-\cot\left(\frac{\pi}{2}\right) - (-\cot\left(\frac{\pi}{4}\right)) = -0 - (-1) = 1$$ So, the definite integral is: $$\int_{\pi / 4}^{\pi / 2} \csc^2 \theta \, d\theta = 1$$

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