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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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Calculus 2 / BC

Chapter 7

Techniques of Integration

Section 1

Integration by Parts

Integration Techniques

Campbell University

Harvey Mudd College

University of Nottingham

Idaho State University

Lectures

01:53

In mathematics, integration is one of the two main operations in calculus, with its inverse, differentiation, being the other. Given a function of a real variable, an antiderivative, integral, or integrand is the function's derivative, with respect to the variable of interest. The integrals of a function are the components of its antiderivative. The definite integral of a function from a to b is the area of the region in the xy-plane that lies between the graph of the function and the x-axis, above the x-axis, or below the x-axis. The indefinite integral of a function is an antiderivative of the function, and can be used to find the original function when given the derivative. The definite integral of a function is a single-valued function on a given interval. It can be computed by evaluating the definite integral of a function at every x in the domain of the function, then adding the results together.

27:53

In mathematics, a technique is a method or formula for solving a problem. Techniques are often used in mathematics, physics, economics, and computer science.

01:48

If $f(0)=g(0)=0$ and $f^{\…

05:17

0:00

If f (0) = g(0) = 0 and f&…

02:22

Show that if $f$ and $g$ a…

04:29

Let f be continuous and g …

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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