Can someone please help me solve these integrals? I've tried to solve them many different ways, but I just can't figure them out.

Question

anonymous · Answer

attachment

anonymous · Answer

Power reduction

anonymous · Answer

What do you mean?

anonymous · Answer

I'm going to have to remember my identities, aren't I?

kirbykirby · Answer

Yes trig identities are key here

anonymous · Answer

Darn. Alrighty, I definitely appreciate you guys pointing that out. I never would have realized. Does that work for the second integral too?

anonymous · Answer

Try using a substitution first, $u=\sin x$, $du=\cos x~dx$, so that you get
$$\int u\sqrt{1+u^2}~du$$
How would you go about this one?

anonymous · Answer

Note: The u-sub isn't actually necessary, you'd find that $u=1+\sin^2x$ also gives you $du=\cos x\sin x~dx$.

kirbykirby · Answer

If you want u could also try substituting $u=\sin^2x+1$ and then $du=2\sin x\cos x$

anonymous · Answer

^^ yup

kirbykirby · Answer

^yes hehe! :)

anonymous · Answer

Wow. I even set the second one up right, but completely failed at something as easy as derivatives. Haha thanks! But I'm still lost on the first one O.o

kirbykirby · Answer

Try using $\sec ^2x=\tan ^2 x+1$

kirbykirby · Answer

like try using $\sec^4x=\sec^2x*\sec^2x$ and substitute that identity into ONE of the $\sec^2x$

kirbykirby · Answer

then you can use $u=\tan x$ so $ du=\sec^2x $

kirbykirby · Answer

you should get some polynomial in terms of $u$'s which is easy to integrate :)

anonymous · Answer

I hate myself so much. Thank you SOOO much for your help!

kirbykirby · Answer

Yw :) But don't say that Integrals aren't always obvious! You really just have to practice them enough to start "seeing" patterns. You'll get more familiar with which techniques might be more useful the more you do them.

kirbykirby · Answer

They're not straightforward like derivatives ! In fact, some integrals cannot be computed by hand and can only be approximated by numerical methods

anonymous · Answer

You're really nice :) Thank you again for all of your help and being so understanding. I just get frustrated because I get so close to the answer and get stuck.

kirbykirby · Answer

Aw thank you :) And you're welcome. The trick is to really do enough of integrals to get used to them ;)