# Thread: Integration help needed, thanks

1. ## Integration help needed, thanks

Hi all,

I'm wondering if someone can point me in the right direction for integrating this integral...

$\int \sin{(\frac{x}{y})} dy$

I'm not sure where to start,

thanks all.

2. Let y = xu. Then it becomes $x\int \sin(\frac1u)du$ which is known not to have an antiderivative amoung the elementary functions. Wolfram Alpha can give it to you with special functions though.

3. If anyone could check through my working it would be great.

Q. Evaluate the following double integral

$\int\int_R \sin{\frac{x}{y}} dA$

Where R is the region bounded by the y-axis, $y = \pi$, $x = y^2$

Drawing a horizontal line throughout the region, we end up with...

$\int^{\pi}_0 \int^{y^2}_0 \sin{\frac{x}{y}} dxdy$

This turns through integration into

$\int^{\pi}_0 [ -y\cos{\frac{x}{y}}]^{y^2}_0 dy$

Subbing in values turns it into

$\int^{\pi}_0 [-y\cos{y} + y] dy$

Integration by parts turning into

$[\frac{y^2}{2} - \cos{y} - y\sin{u}]^{\pi}_0$

Giving

$\frac{\pi^2}{2} - \cos{\pi} - \pi\sin{\pi} + \cos{0}$

Am I right so far?

Thanks.