Page 213 - Calculus Demystified

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EXAMPLE 7.3 CHAPTER 7 Methods of Integration
Calculate

2
log xdx.
1
SOLUTION
This example differs from the previous ones because now we are evalu-
ating a definite integral (i.e., an integral with numerical limits). We still use
the integration by parts formula, keeping track of the numerical limits of
integration.
We first notice that, on the one hand, the integrand is not a product. On the
other hand, we certainly do not know an antiderivative for log x. We remedy
the situation by writing log x = 1 · log x. Now the only reasonable choice is to
take u = log x and dv = 1 dx. Therefore

u(x) = log x du = u (x) dx = (1/x) dx

v(x) = x dv = v (x) dx = 1 dx
and
2 2

1 · log xdx = udv
1 1
2 2

= uv − vdu

1 1
2 2
1
= (log x) · x − x · dx
x
1 1

2
= 2 · log 2 − 1 · log 1 − 1 dx
1
2

= 2 · log 2 − x

1
= 2 · log 2 − (2 − 1)
= 2 · log 2 − 1.
You Try It: Evaluate

4
2
x · sin xdx.
0
We conclude this section by doing another definite integral, but we use a slightly
different approach from that in Example 7.3.