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Suppose that $ f(1) = 2 $, $ f(4) = 7 $, $ f^\pri…

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Problem 70 Hard Difficulty

If $ f(0) = g(0) = 0 $ and $ f^n $ and $ g^n $ are continuous, show that
$$ \int_0^a f(x) g^{\prime\prime} (x) dx = f(a)g^\prime(a) - f^\prime(a)g(a) + \int_0^a f^{\prime\prime} (x) g(x) dx $$

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Chapter 7

Techniques of Integration

Section 1

Integration by Parts

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Video Transcript

This problem is, if f 0 is equal to g 0 is equal to 0 and f to power, and g to power are continuous, show that, integral from 0 to a f times g prime ram dx is equal to f g r, minus f. Prime g s point plus the integral from 0 to a prime prime times g d. For this problem we can't use integration by parts the forener is integral from a to b. U, o d is equal to? U v from a to b minus integral frontin 8 b prim with the ax now for our problem. We can let? U is equal to f of x and the prim is equal to j f. Primpin prom is equal to f prime of x, and v is equal to 2 prim x, so now its integral integral from 0 to a times, g prim prim dx is equal to f g prime from 0 to a minus, integral f prime prime. This is prime d, prime from 0 to 8 d x, so this is equal to a prime, a minus f. 0, since f 0 is 0. So this is minus point and then minus for this integral. We can also use integration by parts. What? U is equal to f, prom and v. Prime is equal to g prime. Then. U, prime, is equal to i, prime prime and v is equal to g. Then this integral is equal to f prime g from 0 to 8 minus the integral from 0 to a pompom times g. So this is equal to f a g prime, a minus. So when we plug in a and 0 to spring, we have. This is f. Prime, a times g a and things g 0 is 0, so this is 0 and the 10 plus integral from 0 to a a prim prim g dthis is just the same way this. So we have proved this equation.

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Calculus: Early Transcendentals

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Problem

Suppose that $ f(1) = 2 $, $ f(4) = 7 $, $ f^\pri…

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Related Courses

Calculus: Early Transcendentals

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Watch More Solved Questions in Chapter 7

Video Transcript

James Stewart

Calculus: Early Transcendentals

Anna Marie Vagnozzi

Kayleah Tsai

Samuel Hannah

Michael Jacobsen

This textbook answer is only visible when subscribed! Please subscribe to view the answer

Calculus: Early Transcendentals

Anna Marie Vagnozzi

Kayleah Tsai

Samuel Hannah

Michael Jacobsen

James StewartCalculus: Early Transcendentals

Anna Marie Vagnozzi

Kayleah Tsai

Samuel Hannah

Michael Jacobsen

James Stewart

Calculus: Early Transcendentals