JEE Main 2020 · 4 September · morning · Single correct · +4 / −1

The integral ∫ (x/(x sin x+cos x))²dx is equal to
(where C is a constant of integration):

  1. (a) sec x-(x tan x)/(x sin x+cos x)+C
  2. (b) sec x+(x tan x)/(x sin x+cos x)+C
  3. (c) tan x-(x sec x)/(x sin x+cos x)+C
  4. (d) tan x+(x sec x)/(x sin x+cos x)+C
Answer and solution

Answer (c) tan x-(x sec x)/(x sin x+cos x)+C

Solution

Maths sir · 4 steps

Notice x cos x is the derivative of the denominator, then split off x sec x and integrate by parts.

  1. Spot the derivative. Let t = x sin x + cos x. Then dt/dx = sin x + x cos x - sin x = x cos x. So split x² as (x sec x)(x cos x) to make dt appear.
  2. Integrate by parts. Take u = x sec x and dv = (x cos x)/t² dx, so v = -1/t. Also du = (sec x + x sec x tan x) dx.

    ∫ x²/t² dx = -(x sec x)/t + ∫ (sec x (1 + x tan x))/(x sin x + cos x) dx

  3. Simplify the remainder. sec x (1 + x tan x) = (cos x + x sin x)/(cos² x). The numerator cancels the denominator, leaving ∫ sec² x dx = tan x.
  4. So. The integral is tan x - (x sec x)/(x sin x + cos x) + C, which is option (c).

The idea to keep. When a squared denominator appears, differentiate it first. Then multiply and divide so that derivative sits on top, and integrate by parts with v = -1/t.

Answer from the NTA final answer key. Solution by PrepWiser.

Study Indefinite Integration

Asked about 60 times in JEE Main since 2020. This question is from 2020.