These freak me out. Haha.
What do I do?
you need to brush up on the properties of the logarithm.
two properties used here (applying them to log base , aka, natural logarithm):
(i)
(ii) or
There are more properties, I HIGHLY RECC' you study them and do problems utilizing them. The premium on knowing the properties of ln, e, as well as those of the core trig functions cannot be overvalued when it comes to calc courses if not the majority of the branches of mathematical sciences!
anyway, we have
=
The last equality is now your integrand. Uber-easy expression to integrate.
sorry just got back, and noticed just now you wanted to differentiate it. as Prove it shows, quite easy. And unfortunately, i just noticed that i made an error above, which is why taking the derivative of the "simplified" form doesnt hold. My error was in claiming:
What i thought i saw when i did that was(the following is true):
=
Also, note every other claim is true. just that one equality, while crucial, was wrong.
sorry for the confusion. first time since ive been posting on this forum that ive made mistakes...let alone 2 in one day! when it rains it pours.