Integrate the following?

Question

Hi,
I need to integrate the following using indefinite integration
I'd be really thankful if any one would be able to help me with any of the question


1] sin4x*cosx*cos6x

2] ( 1 + sin x ) ^ 1/2

3] 1/(sinx*cosx)^2

4] tan x / ( 2(tanx)^2 + 1 )


5] ( 1 + sinx ) / ( 1 + cosx )



Thanks a lot in advance.

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Answer 1

One problem per question please.

3] 1/(sinx*cosx)²

Remember your identities.
sin 2x = 2sinx*cosx

First put the expression into a form more easily integrated.

1/(sinx*cosx)² = 1/[(1/2)(sin 2x)]² = 4/(sin²2x) = 4(csc²2x)

Now we can integrate.

∫[1/(sinx*cosx)²] dx

= ∫[4(csc²2x)]dx = 4(1/2)(-cot 2x) + C

= -2(cot 2x) + C