Integration by Parts Evaluate the integral

[emmaerin]

Homework Statement

Evaluate the integral. (Use C for the constant of integration.)

∫te ^ (-9t) dt

Homework Equations

∫udv = uv - ∫vdu

u=t dv= e ^ (-9t) dt
du=dt v=(-1/9) e ^(-9t)

The Attempt at a Solution

= -1/9 te^(-9t) - ∫-1/9 e ^(-9t) dt

Second Integral:
w=-9t
dw=-9dt
-81∫-1/9 * -81 e ^(-9t) dt
-81∫e^w * w
-81 * e^(-9t) +C

Final Answer:
= -1/9 te^(-9t) + 81 e ^(-9t) +C

This answer isn't right and I'm not sure where I'm going wrong, so any help would be appreciated. Thanks!

 

[Sir Beaver]

A tip which has helped me is to always extract as much as possible before doing the integral. Doing that, you can see that it is actually the same integral as before, e^-9t. The error is simply that you multiplied with 81,while you should have divided.

 

[HallsofIvy]

Homework Statement

Evaluate the integral. (Use C for the constant of integration.)

∫te ^ (-9t) dt

Homework Equations

∫udv = uv - ∫vdu

u=t dv= (-1/9) e ^(-9t)
du=dt v=e ^ (-9t) dt

Where did that "-1/9" come from in "dv"?
The integral you are given is \int te^{-9t}dt. Writing that as \int u dv, you could take u= t, dv= e^{-9t}.

Or you could write the integral as -9\int t(-(1/9)e^{-9t})dt and then take dv= (-1/9)e^{-9t}dt but you don't seem to have done that.

The Attempt at a Solution

= -1/9 te^(-9t) - ∫-1/9 e ^(-9t) dt

Second Integral:
w=-9t
dw=-9dt
-81∫-1/9 * -81 e ^(-9t) dt
-81∫e^w * w
-81 * e^(-9t) +C

Final Answer:
= -1/9 te^(-9t) + 81 e ^(-9t) +C

This answer isn't right and I'm not sure where I'm going wrong, so any help would be appreciated. Thanks!

 

[emmaerin]

Where did that "-1/9" come from in "dv"?

You're completely right - I accidentally switched v and dv, thanks for pointing that out!

 

[emmaerin]

A tip which has helped me is to always extract as much as possible before doing the integral. Doing that, you can see that it is actually the same integral as before, e^-9t. The error is simply that you multiplied with 81,while you should have divided.

So is the final answer -1/9 t e ^(-9t) + 1/81 e ^(-9t) + C ?

 

[LCKurtz]

So is the final answer -1/9 t e ^(-9t) + 1/81 e ^(-9t) + C ?

Differentiate it and see if you get your integrand.