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The second useful tool is the backward version of the product rule. The product rule, as we have noted often, tells us
This means that if we seek an anti-derivative of h(x) and
we can write h as fg', then we can write fg' as (fg)' - f
'g, and an anti-derivative of fg' is then the difference between
any anti-derivative of (fg)' and one of f 'g. But an anti-derivative
of (fg)' is given by fg; so we can use the product rule here
to reduce the problem of finding an anti-derivative of fg'
to finding an anti-derivative of f 'g, for any f and g. This tool is useful for finding anti-derivatives of products
of the form A(x)x if you know an anti-derivative B(x) for
A(x) and an anti-derivative C(x) for B(x) as well. We can
set f = x and g = B in the identity above, and write A(x)x
= B'(x) x, which by this procedure is (Bx)' - B, where we
have used the identity x' = 1. This has Bx - C as an anti-derivative
and Bx- C is therefore an anti-derivative of Ax. The procedure is called integration by parts. It is useful for finding anti-derivatives of products of exponentials and powers or of trigonometric functions and powers or of logarithms and powers, among other things. Exercises:Try integrating the following integrands with respect to x by using this technique: 19.1. x4(ln x) 19.2. x sin x 19.3. x exp x 19.4. (sin x) exp x hint: do it twice and solve the resulting equation. 19.5. x (sin x) exp x (if you can do this you can do them
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