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Question

integrate tan^7(x/2) sec²(x/2)dx

Answer 1

this has a surprisingly simple answer

the trick is to notice that you have a function of tan multiplied by sec^2...and this is important because

d/dx(tanx)=sec^2(x)

so, set u=tan(x/2)

du=1/2 sec^2[x/2] dx

or dx=2 du/sec^2(x/2)

your integral becomes

2 Int(u^7du)

= 2/8 u^8 =

1/4 tan^8(x/2)