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Hey y'all. I'm new to the forum, and have a problem that I've been working on all night long. I'm having issues previewing the Latex, so bear with me. I'll post the work I've done so far if the problem code shows up. Thanks.

[tex]\int{\sin^{\frac{3}{2}}2\theta\cos^{3}2\theta} d\theta[/tex]

Now, the fraction power in the sin is really throwing me off. I've tried rewriting the problem like this:

[tex]\int{\sqrt{\sin^{3}2\theta}\cos^{3}2\theta} d\theta[/tex]

Singled out the sine in the square root:

[tex]\int{\sqrt{\sin2\theta\sin^{2}2\theta}\cos^{3}2\theta} d\theta[/tex]

Square rooted the squared sine function:

[tex]\int{\sqrt{\sin2\theta}\sin2\theta\cos^{3}2\theta} d\theta[/tex]

Skipping some steps (summarized): I split the cosine into two parts ([tex]\cos^{2}2\theta \cos2\theta[/tex]) and then used the trig identity to change [tex]\cos^{2}2\theta[/tex] to [tex](1-\sin^{2}2\theta)[/tex]. Put everything together and got this:

[tex]\int{\sqrt{\sin2\theta}\sin2\theta\cos2\theta-sin^{3}2\theta\cos2\theta d\theta}[/tex]

And now I'm completely lost. I'm not sure if I'm moving in the right direction or just making it a more difficult problem then it really is.

Now that I think about it, I'm not sure if I use integration by parts here.

[tex]\int{\sin^{\frac{3}{2}}2\theta\cos^{3}2\theta} d\theta[/tex]

Now, the fraction power in the sin is really throwing me off. I've tried rewriting the problem like this:

[tex]\int{\sqrt{\sin^{3}2\theta}\cos^{3}2\theta} d\theta[/tex]

Singled out the sine in the square root:

[tex]\int{\sqrt{\sin2\theta\sin^{2}2\theta}\cos^{3}2\theta} d\theta[/tex]

Square rooted the squared sine function:

[tex]\int{\sqrt{\sin2\theta}\sin2\theta\cos^{3}2\theta} d\theta[/tex]

Skipping some steps (summarized): I split the cosine into two parts ([tex]\cos^{2}2\theta \cos2\theta[/tex]) and then used the trig identity to change [tex]\cos^{2}2\theta[/tex] to [tex](1-\sin^{2}2\theta)[/tex]. Put everything together and got this:

[tex]\int{\sqrt{\sin2\theta}\sin2\theta\cos2\theta-sin^{3}2\theta\cos2\theta d\theta}[/tex]

And now I'm completely lost. I'm not sure if I'm moving in the right direction or just making it a more difficult problem then it really is.

Now that I think about it, I'm not sure if I use integration by parts here.

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