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Problem 71 Hard Difficulty

Suppose that $ f(1) = 2 $, $ f(4) = 7 $, $ f^\prime(1) = 5 $, $ f^\prime(4) = 3 $ and $ f^{\prime\prime} $ is continuous. Find the value of $ \displaystyle \int_1^4 x f^{\prime\prime} (x)\ dx $.

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2

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JI

Juste I.

June 21, 2021

Suppose that f(1) = 2, f(4) = 8, f '(1) = 7, f '(4) = 4, and f '' is continuous. Find the value of 4 xf ''(x) dx 1 .

Video Transcript

to evaluate the definite integral from 1 to 4 of x times F double prime of x dx. Given the information here, we would apply integration by parts and here we let u equal to X. And devi would be F double prime of X dx. And so the differential of you would be equal to dx And we would be the integral of deVI which is F prime of X. and so the integral from 1 to 4 of X times F double prime of x dx. This is just U v minus the integral of VD you. This would be extends F prime of x minus the integral of F prime of X. The X The first term evaluated from 1 to 4 And we have this integral from 1-4, integrating further, we would have X times F prime of X Evaluated from 1 to 4 -4 f of X, Evaluated from 1- four. Now we can combine these two and you would get x times f prime of x minus f of X, Evaluated from 1- four. When access for we have four times Ephraim A four -F of four and then when exist one, we would have one times F of one or f rank of one -F of one. This is equal to four times F. Prima for which is equal to three minus Fo four, which is seven and then minus. We have one times F prime of one, which is five -F of one, which is equal to two. Simplifying this, you would get 12 -7 -3 which is equal to two and so this is the value of the definite integral.