Young Jimmy
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Step by step, could somebody break down how to integrate Sec^2(2x)tan^2(2x)

If the solution requires integration by parts or substitution, please explain why you chose that method over the other.

The answer in the mark scheme is 1/6 * tan^3(2x)

thank you!
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_gcx
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Try using the substitution u=tan(2x). (the rest should be relatively straightforward)

Why would I use this method? Because the substitution sticks out at me. (also quicker and less messy)
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Pangol
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i would notice that if you write this as sec^2(2x) (tan(2x))^2, then the first part is the bit in the bracket in the second part differentiated (up to a constant). This means that you can do it by inspection (sometimes called the reverse chain rule in this case).

If you want to do it formally, use the substitution u = tan(2x).

I'm afraid that the best answer I can give to "why this way?" is experience. When you do a lot of these, you'll start to notice the patterns, and the ones that come up often.
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Young Jimmy
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Got it, thanks

Actually guys, I'm still struggling, this is for only 3/75 marks so I don't think a long solution is the right way. It's CCEA exam board and they may have made a mistake. I've checked the formula sheet and only the tan to sec^2x when differentiated identity is provided. Please help!

Integration by Inspection is not on the specification, but I will learn it anyway for these sticky situations
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LikeClockwork
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_gcx
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(Original post by Young Jimmy)
Got it, thanks

Actually guys, I'm still struggling, this is for only 3/75 marks so I don't think a long solution is the right way. It's CCEA exam board and they may have made a mistake. I've checked the formula sheet and only the tan to sec^2x when differentiated identity is provided. Please help!

Integration by Inspection is not on the specification, but I will learn it anyway for these sticky situations
Start by differentiating your substitution to obtain an expression for dx in terms of du, and then adapt the integrand appropriately. You should be left with an integral that will look nice and familiar Note that \frac{d}{dx} tan(kx) = ksec^2(kx), as should be in your formula booklet.
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