## richyw 2 years ago question from first year calculus. Can someone remind me the best way to evaluate $\int\frac{x}{x^2+y^2}\,dy$

1. zepdrix

You're integrating with respect to y? So we treat x as constant, hmm it looks a lot like arctangent. Make this substitution, $\large \color{blue}{y=x \tan \theta}, \quad \color{orangered}{dy=xsec^2 \theta d \theta}$ $\large x \int\limits\limits \frac{dy}{x^2+y^2} \quad = \quad x \int\limits\limits \frac{\color{orangered}{x \sec^2 \theta d \theta}}{x^2+(\color{blue}{xtan \theta)^2}}$

2. zepdrix

Confused? :o If you realize that x is just constant i this case, it might help see what's going on :D

3. richyw

I can see that$\int\frac{\sec^2(\theta)}{1+\tan^2{\theta}}\,d\theta=\theta=\tan^{-1}\frac{y}{x}$

4. richyw

but my question is how do I recognize when to make that substitution? is there anyway to work it out without memorizing that form?

5. richyw

actually never mind. I see how easy it is to recognize this. thanks a lot! (wish we had a formula sheet for multivariable calc haha)

6. zepdrix

Hah :D