luckjoe

Please help the solve the following question
I have tried it for several time but I still can't find the model answer
Here are the question and working

Thank you very much

風之男

Be careful when you do the actual integrate part.
You're integrating over theta, not x, so all the x's in your last steps should be theta.

How do you change back from theta to x? Well, x=3sin theta, so theta = arcsin x/3.
Can you finish now? (It's quite a slog to go from sin4theta to x's using trig identities, but try it first.)

luckjoe

Thank you very much for your sincere attention.
I can’t  modify sin 4 theta to the form of the answer.
Would you give me more hints?
I think I am not smart enough

妙妙貓

I'm not too sure either, but there's an error in your working out; sin^2(2theta) = (1-cos4theta)/2 not (1+cos4theta)/2. So you should end up with a positive (sin4theta)/4 in your answer.

Also, just as a suggestion, if you rearrange your x/3 = sin(theta), you can get theta = arcsin(x/3).

But I'm not sure what to do with the sin4theta

風之男

Don't say you're not smart! You just don't know how to do it _yet_!

So your goal is to express sin(4theta) in terms of sin(theta) and cos(theta).
I'll start you off with sin(4theta) = 2sin(2theta)cos(2theta), can you continue?

Now you might be asking why cos(theta)? Well, that's something you'd do quite a bit in trig. sub - draw a right triangle to find the "missing" length. There should be examples in your textbook.

妙妙貓

Actually, expand the sin(4theta) using double angle formula, then use pythagoras' theorem to find cos(arcsinx/3) and manipulate it algebraically.

luckjoe

Thank you very much for helping me to find the answer.
I’m so delighted that I have finally found the answer.

luckjoe

Thank you very much for your sincere helping