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Use the result of Exercise 27 and the fact that $ \displaystyle \int^{\pi/2}_0 \cos x \, dx = 1 $ (from Exercise 5.1.31), together with the properties of integrals, to evaluate $ \displaystyle \int^{\pi/2}_0 ( 2\cos x - 5x) \, dx $.

$2-\frac{5 \pi^{2}}{8}$

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So in this problem, number 46 were asked to evaluate the definite integral from zero to pi over two to cosign x minus five X. And we haven't got a lot of rules for doing this as of yet. That probably comes in the next section. But they give us two facts that the integral from zero to pirate two of coastline is one. And then a formula for how to integrate the integral from A to B of X. Okay, so I can split this up into Property is integral as this is going to be two times the integral from 0 to Pi over two. Co sign of X. D x minus five. The integral from zero. Excuse me. Yeah, zero to pi over two. Wow, X. Dx. And now I just use these two results that we have. We know that this value right here according to the result, this value Is going to be one. So this is going to be too times one minus five. And then use the result here. The integral from A to B of X. Dx is b squared minus A squared over two. So that's going to be however, two squared minus zero squared over two. Yeah, so this is going to be too minus, that's what pi squared over 4/2. So the minus five hi squared over eight. Which is yeah. Yeah. Yeah. If we move this in terms of eight um No I think that's the final answer. Two um minus five pi squared over eight. I mean what you could do is get a common denominator here of eight And that would leave you what 16 five times square. So either way, so to minus five pi squared over eight.

Florida State University