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Problem 68 Hard Difficulty

If $ \displaystyle f(x) = \int^{\sin x}_0 \sqrt{1 + t^2} \, dt $ and $ \displaystyle g(y) = \int^y_3 f(x) \, dx $, find $ g''(\pi/6) $.

Answer

$$
\frac{\sqrt{15}}{4}
$$

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nm

Nancy M.

September 6, 2020

I'm wondering how did you get the f(y)=h(siny)

Video Transcript

talk about this question uh We are given a function uh in terms of uh different integration and another function G. Y. In terms of the different integration we need to find G double lash by over six. So we're going to use the fundamental theorem of calculus here. And if you find if you are doing G. Dash wire so that's gonna be uh F. F. Y. And the over A. D. Over dx of Y. D. Over dx of why? So that's something but uh that's something but it is gonna be D. U. I. Or D. X. So that is the G. Dash wife. And uh now we need to find the value of uh F. Dash X. Um Actually not that won't be diva already X. Because we are differentiating with respect to y. We're differentiating with respect to Y. So it will just be F. Y. Because diva or diva, it's just gonna be one. And if we take G double dash wire, that's gonna be F dash y. So in short we need to find every dash Y. And that's where we need to look into this equation. So if you look at the the equation than if any of you differentiate that, then F dash X. Again using the phenomenal theorem of calculus, that's going to be square root or one plus sine square X. Times the over the eggs of cynics. So that's going to be in square root of one plus sine square X. And D. Over the eggs. A cynic's Cossacks. So this is a required value of F dash X. But we need to find the value of G double dash pi over six or D. Double dash pi over six is gonna be every dash pi over six and we already have the value of F. Dash X. So every dash pi over six is just by replacing X by pi over six. So it's gonna be one plus sine square pi over six times cause of by over six signed by over six is one or 21 or two squares 1/4 and caused by over 60 through 3/2. So this is gonna be a square root of 5/2 times square root 3/2. which is going to be square root 15 or two which is the because five times three is 15 on two times two is four. So that's going to be square root 15/4 which is the final answer. So G double lash by over six square root of 15/4. Thank you.