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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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  1. zepdrix Group Title
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    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
  2. zepdrix Group Title
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    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
  3. richyw Group Title
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    I can see that\[\int\frac{\sec^2(\theta)}{1+\tan^2{\theta}}\,d\theta=\theta=\tan^{-1}\frac{y}{x}\]

    • one year ago
  4. richyw Group Title
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    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
  5. richyw Group Title
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    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
  6. zepdrix Group Title
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    Hah :D

    • one year ago
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