Integration by Parts

From: Joanne Spiers
Date: 2 August 1999
Subject: Integration by Parts

Integrate e^(2x)*cos(x) from x=0.5 to x=1.5

I have tried integration by parts, but I can't do it.

Maths Help suggests:

You were right to choose to integrate by parts.

We will choose u = e^2x and v' = cos x.
So u' = 2e^2x and v = sin x:

Integrating by parts (*)

You probably had a result similar to the above. The new integral does not look
much simpler than the one you started with. Perhaps you gave up because of this?

But let us continue with this new integral:

We will choose U = e^2x and V' = sin x.
So U' = 2e^2x and V = -cos x:

And substituting this result back into (*) gives:

Notice the integral on the right is the same as the one you started with.

The answer to your integral is 3.361 to 4 significant figures.

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