# Evaluate the following double integral

1. May 18, 2005

### brendan_foo

Just had an exam and I had to evaluate the following double integral, with limited success

$$\int_0^1 \int_0^{\pi} y\sin(xy) {dy} {dx}$$

I managed to compute the first integral, that was ok, using parts. But trying to integrate that with respect to dx just yielded a whole lot of trouble. Could someone have a skim over this and see if its do-able using elementary calculus procedures {I say elementary, but you know what I mean}.

Thankyou

2. May 18, 2005

### Crosson

I would interchange the order of integration and integrate y Sin(x y) with respect to x to get -Cos(x y). Should be easy from here.

3. May 18, 2005

### brendan_foo

Rotation of variables was not covered at all... So say you weren't armed with that tool, what then? (not a cop out, honestly)

Last edited: May 18, 2005
4. May 18, 2005

### dextercioby

Part integration wrt "y" and then integrate the result wrt "x",what else...?

Daniel.

5. May 18, 2005

### brendan_foo

The second integration w.r.t x wouldnt work for me.... Right im going to write down what I've done.

For the first inner integration, i have as follows:

$$\int_0^{\pi} y\cdot \sin(xy) dy = -\frac{\pi}{x}\cos({\pi x}) + \frac{1}{x^2} \cdot \sin({\pi x})$$

I have tried with an abundance of attempts to evaluate all that as an integral with respect to x and I go nowhere conclusive, and its really starting to irritate me. Man, i suck at this.