Math Help - Need Help With Indefinite Integral

This particular integral is driving me crazy. I can't figure it out for the life of me. Could someone please explain to me how to go about solving it?

Integrate[5x(7x - 9)^14dx

I was given a few hints but they still are not helping me at all. First let U=7x-9 and then let x=(U+9)/7. I would imagine this should be done using the substitution method. Thanks in advance

John

Originally Posted by TreeMoney

This particular integral is driving me crazy. I can't figure it out for the life of me. Could someone please explain to me how to go about solving it?

Integrate[5x(7x - 9)^14dx

I was given a few hints but they still are not helping me at all. First let U=7x-9 and then let x=(U+9)/7. I would imagine this should be done using the substitution method. Thanks in advance

John

Is that
INT.[5x (7x -9)^14]dx ?

If yes, then here is one way of doing it.

Integration by parts:
INT.[u]dv = u*v -INT.[v]du ------***

Let u = 5x
So, du = 5dx

And dv = (7x -9)^14 dx = (7x-9)^14 *(7dx /7) = (1/7)(7x-9)^14 *(7dx)
So, v = (1/7)[(1/15)(7x-9)^15] = (1/105)(7x-9)^15

Hence,
INT.[5x (7x-9)^14]dx
= (5x)(1/105)(7x-9)^15 -INT.[(1/105)(7x-9)^15](5dx)
= (1/21)x(7x-9)^15 -(1/21)INT[(7x-9)^15]dx
= (1/21)x(7x-9)^15 -(1/21){(1/7)INT[(7x-9)^15](7dx)}
= (1/21)x(7x-9)^15 -(1/147){(1/16)(7x-9)^16}
= (1/21)x(7x-9)^15 -(1/2352)(7x-9)^16 -------------answer.

Check,
d/dx {(1/21)x(7x-9)^15 -(1/2352)(7x-9)^16}
= (1/21){[x*15(7x-9)^14 *7] +[(7x-9)^15 *1]} -(1/2352){16(7x-9)^15 *7}
= {5x(7x-9)^14 +(1/21)(7x-9)^15} -(1/21)(7x-9)^15
= 5x(7x-9)^14
Okay.

Originally Posted by TreeMoney

This particular integral is driving me crazy. I can't figure it out for the life of me. Could someone please explain to me how to go about solving it?

Integrate[5x(7x - 9)^14dx

I was given a few hints but they still are not helping me at all. First let U=7x-9 and then let x=(U+9)/7. I would imagine this should be done using the substitution method. Thanks in advance

John

It can be done by integration by parts, but also directly by substitution.






Which you can then simplify any way you wish.

-Dan

Thanks Guys. Now that I see how it is done, I was making stupid algebra mistakes. Thanks again for the help. It's much appreciated.

John

Originally Posted by TreeMoney

Thanks Guys. Now that I see how it is done, I was making stupid algebra mistakes. Thanks again for the help. It's much appreciated.

John

There's no such thing as a "stupid mistake." You either get it right or you don't. You didn't get it right, you knew you didn't, and you asked a question about it. I see no stupidity there, I see a good practice in action!

-Dan