Integration by parts (example 2)

Q:  ∫(y)/e^(2y) from 0 to 1.

A:  Integration by parts it is!

∫udv = uv – ∫vdu

u = y                   v = – 1/2 e^(-2y)

du = dy              dv = e^(-2y)

So, ∫(y)/e^(2y) = -(1/2) y e^(-2y) – ∫ -(1/2) e^(-2y) dy

With house-keeping gives:

-(1/2) y e^(-2y) + 1/2 ∫ e^(-2y) dy

And integrate the last piece to get:

-(1/2) y e^(-2y) – 1/4 e^(-2y)

Now, from o to 1:

[-(1/2) *1* e^(-2*1) – 1/4 e^(-2*1)] – [-(1/2) *0* e^(-2*0) – 1/4 e^(-2*0)]

With careful simplification gives:

– 3/4 e^(-2) + 1/4

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Google photo

You are commenting using your Google account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s