Question

What is f(x)=∫cot2xdx if f(π8)=0?

Answer 1

Answer.......

f(x)=1/2ln√2sin(2x)

Explanation:

f(x)=∫cot2xdx

f(x)=∫(cos2x/sin2x)dx

now

(d/dx(sin2x)=2cos2x)

we have a log integration.

f(x)=∫(cos2x/sin2x)dx=1/2lnsin2x+C

f(π/8)=0

∴1/2lnsin(π/4)+C=0

rewrite C=1/2lnK in order to include it in the log.

1/2lnsin(π/4)+1/2lnk=0

1/2[(lnsin(π/4)+lnk]=0

1/2(lnksin(π/4))=0

lnx=0⇒x=1 ∴ksin(π/4)=1

k(√2/2)=1

k=2/√2=√2

f(x)=1/2ln√2sin(2x)