Evaluate
In this question, first of all we expand cot4x as
cot4x = (cosec2x – 1)2
= cosec4x – 2cosec2x + 1 …(1)
Now, write cosec4x as
cosec4x = cosec2xcosec2x
= cosec2x(1 + cot2x)
= cosec2x + cosec2xcot2x
Putting the value of cosec4x in eq(1)
cot4x = cosec2x + cosec2xcot2x – 2cosec2x + 1
= cosec2xcot2x – cosec2x + 1
y = ∫ cot4x dx
= ∫ cosec2x cot2x dx + ∫ - cosec2x + 1 dx
A = ∫cosec2x cot2x dx
Let, cot x = t
Differentiating both side with respect to x
⇒ -dt = cosec2x dx
= ∫-t2 dt
Using formula
Again, put t = cot x
Now, B= ∫-cosec2 x +1 dx
Using formula ∫cosec2 x dx= -cot x and ∫c dx=cx
B = cot x + x + c2
Now, the complete solution is
y = A + B