Q20 / 104 · NTA Abhyas (JEE Main) 2020 · Mathematics · Indefinite Integration

The value of ∫cos⁡xsin2⁡xsin⁡x+cos⁡xdx is equal to

  • Correct answer
    A
    log⁡1+tan⁡xtan⁡x-cot⁡x+C
  • B
    log⁡tan⁡x1+tan⁡x+C
  • C
    log⁡tan⁡x1+tan⁡x-tan⁡x+C
  • D
    log⁡tan⁡x1+tan⁡x+cot⁡x+C

Solution

Let, I= ∫cos⁡xsin2⁡xsin⁡x+cos⁡xdx

On dividing numerator and denominator by cos3⁡x, and substituting tanx=t, sec2xdx=dt, we get,

I= ∫dtt21+t

= ∫-1t+1t2+11+tdt (using partial fraction)

⇒I=− ∫ 1 t dt + ∫ t −2 dt+ ∫ 1 1+t dt

=-log⁡t-1t+log⁡1+t+C

= -1tan⁡x+log⁡1+tan⁡xtan⁡x+C

=-cot⁡x+log⁡1+tan⁡xtan⁡x+C

Hence, the correct answer is option: 'A'.

More Indefinite Integration Questions