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edexcel FP3 integration - June 2012 - Q7c

how does the integral of 2e^x / (e^2x +9) = 2[ arctan(e^x/3) ] /3I thought it would be 2e^x [arctan(e^x /3)]/3, as the general expression for integrating 1/a^2+x^2 = [arctan(x/a)] / aso seeing as there is an e^x in the numerator, would it not be what i wrote above in the 2nd line?Need help urgently as the exam is TOMORROW MORNING, so I appreciate all full solutions or comments.THANKS IN ADVANCE!! :biggrin:
Original post by Coral Reafs
how does the integral of 2e^x / (e^2x +9) = 2[ arctan(e^x/3) ] /3I thought it would be 2e^x [arctan(e^x /3)]/3, as the general expression for integrating 1/a^2+x^2 = [arctan(x/a)] / aso seeing as there is an e^x in the numerator, would it not be what i wrote above in the 2nd line?Need help urgently as the exam is TOMORROW MORNING, so I appreciate all full solutions or comments.THANKS IN ADVANCE!! :biggrin:

You haven't integrated by substitution correctly. Remember, when you integrate by substitution using u(x)u(x), you need to remember to convert dxdu\text{d}x \mapsto \text{d}u by multiplying the integrand by dudx\dfrac{du}{dx}.
Original post by Coral Reafs
how does the integral of 2e^x / (e^2x +9) = 2[ arctan(e^x/3) ] /3I thought it would be 2e^x [arctan(e^x /3)]/3, as the general expression for integrating 1/a^2+x^2 = [arctan(x/a)] / aso seeing as there is an e^x in the numerator, would it not be what i wrote above in the 2nd line?Need help urgently as the exam is TOMORROW MORNING, so I appreciate all full solutions or comments.THANKS IN ADVANCE!! :biggrin:


Chain rule.

What's the derivative of e^x?
Original post by Smaug123
You haven't integrated by substitution correctly. Remember, when you integrate by substitution using u(x)u(x), you need to remember to convert dxdu\text{d}x \mapsto \text{d}u by multiplying the integrand by dudx\dfrac{du}{dx}.


Original post by alexmufc1995
Chain rule.

What's the derivative of e^x?


Oh yeah right, u=e^x, du/dx = e^x, so dx = du/e^x

so expression becomes integral of --> 2/u^2 + 9, so you get (2/3)arctan( u/3)

= > (2/3)arctan(e^x /3)??

Is this correct?

btw many thanks for both of your quick contributions at this crucial time, +1 if it means anything.
Original post by Coral Reafs
Oh yeah right, u=e^x, du/dx = e^x, so dx = du/e^x

so expression becomes integral of --> 2/u^2 + 9, so you get (2/3)arctan( u/3)

= > (2/3)arctan(e^x /3)??

Is this correct?

btw many thanks for both of your quick contributions at this crucial time, +1 if it means anything.

Yep, that's right :smile: thanks!

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