Page 210 - Calculus Demystified
P. 210
CHAPTER 7
Methods of
Integration
7.1 Integration by Parts
We learned in Section 4.5 that the integral of the sum of two functions is the sum
of the respective integrals. But what of the integral of a product? The following
reasoning is incorrect:
2
x dx = x · xdx = xdx · xdx
3 2 2
because the left-hand side is x /3 while the right-hand side is (x /2) · (x /2) =
4
x /4.
The correct technique for handling the integral of a product is a bit more subtle,
and is called integration by parts. It is based on the product rule
(u · v) = u · v + u · v .
Integrating both sides of this equation, we have
(u · v) dx = u · vdx + u · v dx.
The Fundamental Theorem of Calculus tells us that the left-hand side is u · v. Thus
u · v = u · vdx + u · v dx
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