Perhaps with just a little more experience, you will see that this one is screaming for a substitution. Try u = e^(6x) + 10.
Can someone please help me?
find the exact value of:
ln7
⌠
⌡ int e^(6x)(e^(6x) + 10)^7/2 dx
ln5
I have a brief idea of how to solve this, but I'm not too sure about the specific parts.
You can make e^6x=u, differentiate this to find dx, but I'm not too sure about the new limits and how to solve the rest. Is there any easier way to solve this, other than the substitution method?
any help would be great, thanks
Hi!
I remember at school there used to be this formula that we blindly used as a tool whenever something like this popped up, then we learnt that it's in fact a form substitution method:
.
Can you know why? Just remember that
And remember, by this so called change of variable method we're now integrating w.r.t. u and not x. So if and the limits are x = 0, 1, the limits of u are 1, 3.
That plus TKHunny's hint, can you complete this?
Let u = e^{6x} + 10. Your lower limit for x is ln(5), so plug x = ln(5) into the u formula to get the lower limit for u...
For the rest of it, you can easily solve for e^{6x} from the u equation. So you can find the integrand in terms of u. Finally you can find du = ( ) dx. Solve for dx and plug that into the integral. You will get
Can you take it from there?
-Dan
Hello all!
Oh dear, topsquark, I don't know why you want to go there! Since isn't just any regular function, why not going as follows:
and noticing that . Then:
= between your 2 modified limits.
Other version:
⌠
⌡(1/6). u'(x) . u^n(x) dx =
⌠
⌡(1/6) u^(7/2) du = (1/6) / (9/2) u ^ (9/2) = (1/27) . (e^(6x) + 10) ^ (9/2)
Somebody please do something with the LaTeX compiler!
(shrugs) It's not like it's that hard. And it's a simple and direct application of the substitution method. But you like your way and I like mine. No problems.
See post 8 in this thread.
-Dan