/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 44 Evaluate the following integrals... [FREE SOLUTION] | 91影视

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Chapter 5: Problem 44

Evaluate the following integrals using the Fundamental Theorem of Calculus. $$\int_{-\pi / 2}^{\pi / 2}(\cos x-1) d x$$

Short Answer

Expert verified
Question: Evaluate the definite integral of the function \((\cos x - 1)\) from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\). Answer: The definite integral is equal to \(2 - \pi\).

Step by step solution

01

Applying the Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus states that if \(f(x)\) is a continuous function on an interval \([a, b]\), then: $$\int_{a}^{b} f(x)dx = F(b) - F(a)$$ where \(F(x)\) is an antiderivative of \(f(x)\). In this case, \(f(x) = (\cos x - 1)\), and the interval is from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\). Let's find the antiderivative \(F(x)\).
02

Finding the antiderivative of \(f(x)\)

We are given \(f(x) = (\cos x - 1)\), so we need to find its antiderivative, \(F(x)\). The antiderivative of \(\cos x\) is \(\sin x\), since the derivative of \(\sin x\) is \(\cos x\). The antiderivative of a constant (in this case, \(-1\)) is a linear term with the constant as the coefficient. Therefore, the antiderivative of \(-1\) with respect to \(x\) is \(-x\). So the antiderivative \(F(x)\) is: $$F(x) = \sin x - x$$
03

Applying the Fundamental Theorem of Calculus

Now that we have the antiderivative \(F(x)\), we can apply the Fundamental Theorem of Calculus to evaluate the definite integral: $$\int_{-\pi / 2}^{\pi / 2}(\cos x-1)d x = F(\pi/2) - F(-\pi/2)$$
04

Evaluating the antiderivative function in the endpoints

To evaluate the difference \(F(\pi/2) - F(-\pi/2)\), we substitute the interval's endpoints in \(F(x)\): $$F(\pi/2) = \sin(\pi/2) - (\pi/2) = 1-\pi/2$$ $$F(-\pi/2) = \sin(-\pi/2) - (-\pi/2) = -1+\pi/2$$
05

Calculating the definite integral

Now, we subtract the value of \(F(-\pi/2)\) from \(F(\pi/2)\) to find the definite integral: $$\int_{-\pi / 2}^{\pi / 2}(\cos x-1)d x = (1-\pi/2) - (-1+\pi/2) = (1+1) - \pi = \boxed{2 -\pi}$$

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Most popular questions from this chapter

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