Evaluating Indefinite Integrals for Dan

[danago]

Hey. Say i was given this indefinite integral to evaluate:

http://img126.imageshack.us/img126/6374/aaaaog3.gif

How could i do that? I can do it by first expanding it all, but that takes a very long time and is quite tedious, especially with such a large index as 7. Is there another way i can do that?

Thanks,
Dan.

 

[StatusX]

You could use substitution.

 

[danago]

What would i substitute out? I tried using substitution, but couldn't get very far.

 

[danago]

http://img109.imageshack.us/img109/8527/aaaaac7.gif

Where from there?

 

[HallsofIvy]

Why squared? that leaves a fraction power. I would just let
u= 3x+2 then du= 3dx so dx= (1/3)du. Also, x= u/3+ 2/3 so
x(3x+2)⁷dx= (u/3+ 2/3)(u⁷)(1/3du)= (1/9)(u⁸+ 2u⁷)dx

Danago, you have remember to replace dx with du. If u= (3x+2)², then du= 2(3x+2)dx= (6x+ 4)dx. I think courtrigrad's point was that you can use that "x" in the integeral to help with that. But with that "4dx" still left, I think u= 3x+2 is simpler.

 

[courtrigrad]

doesn't x = u/3 - 2/3

 

[danago]

Why squared? that leaves a fraction power. I would just let
u= 3x+2 then du= 3dx so dx= (1/3)du. Also, x= u/3+ 2/3 so
x(3x+2)⁷dx= (u/3+ 2/3)(u⁷)(1/3du)= (1/9)(u⁸+ 2u⁷)dx

Danago, you have remember to replace dx with du. If u= (3x+2)², then du= 2(3x+2)dx= (6x+ 4)dx. I think courtrigrad's point was that you can use that "x" in the integeral to help with that. But with that "4dx" still left, I think u= 3x+2 is simpler.

Hmmm...im a bit lost. I understand up to "dx= (1/3)du", but where does the "x= u/3+ 2/3" come from? If it is anything more advanced than substitution, i think ill leave it there, because i just made this question up out of curiosity, not because i need to know it for school, and even substitution is more advanced than what we've been doing at school, but i learned it to make some of the question we do simpler.

 

[courtrigrad]

If u = 3x +2, then x = u/3 - 2/3. You have to solve for x.

 

[danago]

oh lol. That simple :P

 

[danago]

Is this right?
http://img150.imageshack.us/img150/1972/aaaaka4.gif

 

[Max Eilerson]

Yes it's correct, but you can tidy it up a little.

 

[danago]

ok thanks everyone :)