Hard definite integration by parts question,

[limelightdevo]

Homework Statement

Integral, from 0 to 1, y/(e^2y)

Homework Equations

Use integration by parts

The Attempt at a Solution

I need integration by parts. The answer is 1/4-3/5e^-2. But I got 1/2 and it all makes sense to me, so tell me what I got wrong.

I put u=e^2y
du=2e^2y

dv= 1/e^2y dy (did u substitution on 2y here)
v=1/2 * ln(e^2y) (here I canceled ln and e, so left with 2y)
v=1/2 * 2y = y

Plugged them into integration by parts formula
and got y^2-(y^2)/2. Plugged in 1 and 0.
And I got 1-1/2 = 1/2. How is this wrong?? I don't understand.

 

[1MileCrash]

Your integration of dv is wrong.

 

[SammyS]

Homework Statement

Integral, from 0 to 1, from y/e^2y

Homework Equations

Use integration by parts

The Attempt at a Solution

I need integration by parts. The answer is 1/4-3/5e^-2. But I got 1/2 and it all makes sense to me, so tell me what I got wrong.

I put u=e^2y
du=2e^2y

dv= 1/e^2y dy (did u substitution on 2y here)
v=1/2 * ln(e^2y) (here I canceled ln and e, so left with 2y)
v=1/2 * 2y = y

Plugged them into integration by parts formula
and got y^2-(y^2)/2. Plugged in 1 and 0.
And I got 1-1/2 = 1/2. How is this wrong?? I don't understand.

What is it that you're trying to integrate? What you wrote is ambiguous.

$\displaystyle \int_0^1 \frac{y}{e^{2y}}\,dy\,,$

or

$\displaystyle \int_0^1 y\,e^{2y}\,dy\,,$

or something else?

 

[Mark44]

Homework Statement

Integral, from 0 to 1, from y/e^2y

Homework Equations

Use integration by parts

The Attempt at a Solution

I need integration by parts. The answer is 1/4-3/5e^-2. But I got 1/2 and it all makes sense to me, so tell me what I got wrong.

I put u=e^2y
du=2e^2y

dv= 1/e^2y dy (did u substitution on 2y here)
v=1/2 * ln(e^2y) (here I canceled ln and e, so left with 2y)
v=1/2 * 2y = y

Plugged them into integration by parts formula
and got y^2-(y^2)/2. Plugged in 1 and 0.
And I got 1-1/2 = 1/2. How is this wrong?? I don't understand.

In addition to what 1MileCrash said, you aren't doing integration by parts correctly. If you integral is $\int ye^{-2y}dy$
then you can't have u = e^2y and dv = 1/(e^2y)* dy.
Whatever you choose for u and dv, they have to multiply to make your original integrand. In your case u*dv = e^2y/e^2y * dy = 1 dy, not ye^-2ydy.

 

[1MileCrash]

So your choice of u and dv should be very clear. As for which one is u and which one is dv - weigh which one will result in a simpler function when differentiated, and try that for u.

 

[limelightdevo]

It is y divided by e^2y dy

 

[Mark44]

It is y divided by e^2y dy

That's what I understood it to be.

You should write this not as a fraction, but as ye^-2ydy.