Assertion (A): if I n = ∫ cot n x d x , then I 6 + I 4 = - cot 5 x 5 Reason (R): ∫ cot n x d x =…

Assertion (A): if In=∫cotnxdx, then I6+I4=-cot5x5

Reason (R): ∫cotnxdx=-cotn-1xn-∫cotn-2xdx

  1. A is false, R is false
  2. A is true, R is true
  3. A is true, R is false
  4. A is false, R is true

Solution

In=∫cotnxdx

=∫cotn-2xcot2xdx

=∫cotn-2xcosec2xdx-∫cotn-2xdx

=I-In-2

Here, I=∫cotn-2xcosec2xdx

Let cotx=t; cosec2xdx=-dt

i.e. I=-∫tn-2dt=tn-11-n=cotn-1x1-n

I.e. In=cotn-1x1-n-In-2

⇒I6+I4=-cot5x5

Hence, assertion is true but reason is false.

 

Asked in: AP EAMCET 2022 (04 Jul Shift 1)

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