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richyw
question from first year calculus. Can someone remind me the best way to evaluate \[\int\frac{x}{x^2+y^2}\,dy\]
You're integrating with respect to y? So we treat x as constant, hmm it looks a lot like arctangent. Make this substitution, \[\large \color{blue}{y=x \tan \theta}, \quad \color{orangered}{dy=xsec^2 \theta d \theta}\] \[\large x \int\limits\limits \frac{dy}{x^2+y^2} \quad = \quad x \int\limits\limits \frac{\color{orangered}{x \sec^2 \theta d \theta}}{x^2+(\color{blue}{xtan \theta)^2}}\]
Confused? :o If you realize that x is just constant i this case, it might help see what's going on :D
I can see that\[\int\frac{\sec^2(\theta)}{1+\tan^2{\theta}}\,d\theta=\theta=\tan^{-1}\frac{y}{x}\]
but my question is how do I recognize when to make that substitution? is there anyway to work it out without memorizing that form?
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)