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Question:

Let α∈(0,π/2) be fixed. If the integral ∫tanx+tanα/(tanx−tanα)dx=A(x)cos2α+B(x)sin2α+C, where C is a constant of integration, then the functions A(x) and B(x) are respectively.

x−αandloge|cos(x−α)

x−αandloge|sin(x−α)

x+αandloge|sin(x+α)

x+αandloge|sin(x−α)

Solution:

Correct option is C.
x−αandloge|sin(x−α)
∫tanx+tanα/(tanx−tanα)dx=∫sin(x+α)/sin(x−α)dx
Let x−α=t ⇒∫sin(t+2α)/sint dt=∫cos2αdt+∫cot(t)sin2αdt=t.cos2α+ln|sint|.sin2α+
=(x−α)cos2α+ln|sin(x−α)|.sin2α+C.