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Calculus/Definite Integral using Substitution

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Expert: - 2/5/2008


Question
I am taking a class over Integral Calculus. Can you please help me with the following problem?
Evaluate the definite integral:
I: [4,12] (X-1)/(x-3)^(1/2) dx

I let u = x-3
     du = 1dx
     x = u + 3
     the new interval for u = [1,9]

Substituting u in the equation, I get
I= [1,9] (u+2)/u^(1/2) du
and when I take the antiderivative, I get
(u^2)/2 + 2u x 2u^(1/2)
When I plug in 9 and 1, I get
351 - 5 = 346
Unfortunately 346 is not an available answer on my homework. Can you please assist me and point out where I may be going wrong?

Answer
You are right up to where you have
I= [1,9] (u+2)/u^(1/2) du

Before doing integration, lets split this into two integrals:
I= [1,9] u/u^(1/2) du  + [1,9] 2/u^(1/2) du.

The first integral is then
I= [1,9] u^(1/2) du
and the second integral is [1,9] 2u^-(1/2) du.

Evaluate these two integrals to get u^(3/2)/(3/2) + 2u(1/2)/(1/2), evaluated from 1 to 9.  Does this work?

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