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Calculus/Trouble with integration

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Question
Hi Amos!

These two problems are giving me very much trouble.

1. find the value of "a" for the integral

(x e^x) dx =3a , lower limit: 1, upper limit: ln (a)

2. Find the value of the integral

e^sqrt(ax) d(sqrt x) , lower limit: 0, upper limit: 4/a

Thank you!

Answer
The second problem is easier, but the first one involves more algebra that respects logarithm.

1)  Assuming you know how to integrate by parts, the first one ha s function whose anti-derivative (or integral) is F(x)= xe^x - e^x.
So the definite integral by the fundamental theorem of calculus becomes F(ln a) - F(1).  i.e. we have the equation:
         F(ln a) - F(1)  = 3a
=>    (ln a -1) a  - (1e^1 - e^1) = 3a
=>    (ln a -1) a   = 3a
=>    (ln a -4) a = 0  (I hope you understand the algebra needed to get here)
so either a = 0  or ln a = 4
i.e a = 0 or a = e^4.
But we want a limit from 1 to ln(a), so a cannot be zero, that leaves us with the asnwer a = e^4.

2)  If you do not understand what d(sqrt x) means, please refer to integration by substitution.
Doing so, we will get the integral of e^sqrt(ax) d(sqrt x)  = G(x) = e^sqrt(ax)  / sqrt(a).

So again using the fundamental theorem of calculus, the definite integral becomes G(4/a) - G(0)

G(4/a) = e^2/sqrt(a)  and G(0) = 1.

Hence the integral e^sqrt(ax) d(sqrt x) , lower limit: 0, upper limit: 4/a is equal to

(e^2 - 1) / sqrt(a)

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Amos

Expertise

I can answer all calculus question. I am a Math Lecturer and I teach Math in a College, usually Calculus 1, Calculus 2 and Calculus 3 and Linear Algebra.

Experience

I have been teaching in this college for more than 12 years. Prior to that, I taught for three years as Visiting Assistant Professor.

Education/Credentials
I got my doctorate from Univ. of Rochester in Algebraic Toppology. I got a MA from Univ of Rochester, an MA from York Univ. in Toronto and almost did my M.Sc in the National Univ. of Singapore.

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