Difference Equations to Differential Equations (24).pdf

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 £© More Techniques of Integration
In the last section we saw how we could exploit our knowledge of the chain rule to develop
a technique for simplifying integrals using suitably chosen substitutions. In this section
we shall see how we can develop a second technique, called integration by parts, using the
product rule. Outside of algebraic manipulation and the use of various functional identities,
like the trigonometric identities, substitution and parts are the only basic techniques we
have available to us for simplifying the process of evaluating an integral.
Example Suppose we wish to find x cos(x)dx. Since
R
Z cos(x)dx = sin(x) + c,
we might make an initial guess of F (x) = x sin(x) for an antiderivative of f(x) = x cos(x).
But, of course, differentiation of F , using the product rule, yields
F 0(x) = x cos(x) + sin(x),
which differs from the desired result, f(x), by the term sin(x). However, since
Z sin(x)dx = − cos(x) + c,
we can obtain an antiderivative of f(x) by adding on the term cos(x) to F (x). That is,
G(x) = x sin(x) + cos(x)
is an antiderivative of f(x) since the derivative of cos(x) will cancel the sin(x) term in
F 0(x). Explicitly,
G0(x) = x cos(x) + sin(x) − sin(x) = x cos(x).
Thus
Z x cos(x)dx = x sin(x) + cos(x) + c.
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