Let I_n = ∫ tan^n x dx (n > 1). If I_4 + I_6 = a tan^5 x +… — JEE Main 2017 Mathematics PYQ

JEE Main 2017

Let

I_n = \int \tan^n x \, dx

(n > 1)

. If

I_4 + I_6 = a \tan^5 x + bx^5 + C

, where

C

is a constant of integration, then the ordered pair

(a, b)

is equal to

A.
$\left(\dfrac{1}{5}, -1\right)$
B.
$\left(\dfrac{1}{5}, 0\right)$
C.
$\left(\dfrac{1}{5}, -1\right)$
D.
$\left(-\dfrac{1}{5}, 0\right)$

Show answer and solution

Correct option: B

Solution

Using the reduction identity: I_n + I_(n+2) = tan^(n+1)(x)/(n+1).

So I₄ + I₆ = tan⁵(x)/5 + C.

Comparing with a·tan⁵(x) + b·x⁵ + C gives a = 1/5 and b = 0.

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JEE Main 2017 Mathematics question on Indefinite Integration