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richyw
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question from first year calculus. Can someone remind me the best way to evaluate \[\int\frac{x}{x^2+y^2}\,dy\]
 one year ago
 one year ago
richyw Group Title
question from first year calculus. Can someone remind me the best way to evaluate \[\int\frac{x}{x^2+y^2}\,dy\]
 one year ago
 one year ago

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zepdrix Group TitleBest ResponseYou've already chosen the best response.1
You're integrating with respect to y? So we treat x as constant, hmm it looks a lot like arctangent. Make this substitution, \[\large \color{blue}{y=x \tan \theta}, \quad \color{orangered}{dy=xsec^2 \theta d \theta}\] \[\large x \int\limits\limits \frac{dy}{x^2+y^2} \quad = \quad x \int\limits\limits \frac{\color{orangered}{x \sec^2 \theta d \theta}}{x^2+(\color{blue}{xtan \theta)^2}}\]
 one year ago

zepdrix Group TitleBest ResponseYou've already chosen the best response.1
Confused? :o If you realize that x is just constant i this case, it might help see what's going on :D
 one year ago

richyw Group TitleBest ResponseYou've already chosen the best response.0
I can see that\[\int\frac{\sec^2(\theta)}{1+\tan^2{\theta}}\,d\theta=\theta=\tan^{1}\frac{y}{x}\]
 one year ago

richyw Group TitleBest ResponseYou've already chosen the best response.0
but my question is how do I recognize when to make that substitution? is there anyway to work it out without memorizing that form?
 one year ago

richyw Group TitleBest ResponseYou've already chosen the best response.0
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)
 one year ago
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