Indefinite IntegrationHard
Question
If In = ∫ cotn x dx and I0 + I1 + 2 (I2 + .....+ I8) + I9 + I10 = A 
+ C, where u = cot x and C is an arbitrary constant, then
+ C, where u = cot x and C is an arbitrary constant, then Options
A.A is constant
B.A = - 1
C.A = 1
D.A is dependent on x
Solution
In =∫cotn x dx = ∫cotn-2x(cos ec2x - 1)dx = 
- In - 2 + C1
⇒ In + In - 2 =
+ C1
Now I0 + I1 + 2 (I2 + .....+ I8) + I9 + I10
= (I2 + I0) + (I3 + I1) + (I4 + I2) + (I5 + I3) + (I6 + I4) + (I7 + I5) + (I8 + I6) + (I9 + I7) + (I10 + I8)
∴ A = - 1
- In - 2 + C1⇒ In + In - 2 =
+ C1Now I0 + I1 + 2 (I2 + .....+ I8) + I9 + I10
= (I2 + I0) + (I3 + I1) + (I4 + I2) + (I5 + I3) + (I6 + I4) + (I7 + I5) + (I8 + I6) + (I9 + I7) + (I10 + I8)
∴ A = - 1
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