AP EAMCET20224 Jul 2022Morning ShiftMathematicsIndefinite IntegrationActual
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 = - cot n - 1 x n - ∫ cot n - 2 x d x
Options
- AA is false, R is false
- BA is true, R is true
- CA is true, R is false
- DA is false, R is true
Correct answer
C. A is true, R is false
Step-by-step solution
I n = ∫ cot n x d x = ∫ cot n - 2 x cot 2 x d x = ∫ cot n - 2 x cosec 2 x d x - ∫ cot n - 2 x d x = I - I n - 2 Here, I = ∫ cot n - 2 x cosec 2 x d x Let cot x = t ;   cosec 2 x d x = - d t i.e. I = - ∫ t n - 2 d t = t n - 1 1 - n = cot n - 1 x 1 - n I.e. I n = cot n - 1 x 1 - n - I n - 2 ⇒ I 6 + I 4 = - cot 5 x 5 Hence, assertion is true but reason is false.