richyw
  • richyw
question from first year calculus. Can someone remind me the best way to evaluate \[\int\frac{x}{x^2+y^2}\,dy\]
Mathematics
chestercat
  • chestercat
I got my questions answered at brainly.com in under 10 minutes. Go to brainly.com now for free help!
At vero eos et accusamus et iusto odio dignissimos ducimus qui blanditiis praesentium voluptatum deleniti atque corrupti quos dolores et quas molestias excepturi sint occaecati cupiditate non provident, similique sunt in culpa qui officia deserunt mollitia animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil impedit quo minus id quod maxime placeat facere possimus, omnis voluptas assumenda est, omnis dolor repellendus. Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis voluptatibus maiores alias consequatur aut perferendis doloribus asperiores repellat.

Get this expert

answer on brainly

SEE EXPERT ANSWER

Get your free account and access expert answers to this
and thousands of other questions

zepdrix
  • zepdrix
You're integrating with respect to y? So we treat x as constant, hmm it looks a lot like arctangent. Make this substitution, \[\large \color{blue}{y=x \tan \theta}, \quad \color{orangered}{dy=xsec^2 \theta d \theta}\] \[\large x \int\limits\limits \frac{dy}{x^2+y^2} \quad = \quad x \int\limits\limits \frac{\color{orangered}{x \sec^2 \theta d \theta}}{x^2+(\color{blue}{xtan \theta)^2}}\]
zepdrix
  • zepdrix
Confused? :o If you realize that x is just constant i this case, it might help see what's going on :D
richyw
  • richyw
I can see that\[\int\frac{\sec^2(\theta)}{1+\tan^2{\theta}}\,d\theta=\theta=\tan^{-1}\frac{y}{x}\]

Looking for something else?

Not the answer you are looking for? Search for more explanations.

More answers

richyw
  • 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?
richyw
  • 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)
zepdrix
  • zepdrix
Hah :D

Looking for something else?

Not the answer you are looking for? Search for more explanations.