Question

integration sin x Cos x/ sin x + cos x dx

Answer 1

Integrand:  f(x) = Sinx Cosx dx / [ sin x + cos x ] 
Answer will be :
   1/2√2  * Ln | (√2 + sinx + cosx)/(sinx+cosx) - 1/2 * (Cosx - Sin x) + K

f(x) = 1/2 * [ (sinx + cosx)² - 1 ] / [ sin x + cos x ]
      = 1/2 * (sin x + cos x) - 1/2 * 1/(sin x + cos x)
      = 1/√2 * Sin(π/4 + x) - 1/2√2 * Cosec (π/4 + x)
                                    as  √2 * Sin(π/4 + x) = sinx + cosx.

We integrate f(x) dx now to get: 
I = int  f(x) dx
  = - 1/√2 * Cos(π/4+ x) + 1/2√2  * Ln | Cosec(π/4) + Cot(π/4 +x) | + K
 =  1/2√2  * Ln | (√2 + sinx + cosx)/(sinx+cosx) - 1/2 * (Cosx - Sin x) + K

Answer 2

Answer:

hey, here is ur answer

Step-by-step explanation:

Integrand:  f(x) = Sinx Cosx dx / [ sin x + cos x ] 

Answer will be :

   1/2√2  * Ln | (√2 + sinx + cosx)/(sinx+cosx) - 1/2 * (Cosx - Sin x) + K

f(x) = 1/2 * [ (sinx + cosx)² - 1 ] / [ sin x + cos x ]

      = 1/2 * (sin x + cos x) - 1/2 * 1/(sin x + cos x)

      = 1/√2 * Sin(π/4 + x) - 1/2√2 * Cosec (π/4 + x)

                                    as  √2 * Sin(π/4 + x) = sinx + cosx.

We integrate f(x) dx now to get: 

I = int  f(x) dx

  = - 1/√2 * Cos(π/4+ x) + 1/2√2  * Ln | Cosec(π/4) + Cot(π/4 +x) | + K

 =  1/2√2  * Ln | (√2 + sinx + cosx)/(sinx+cosx) - 1/2 * (Cosx - Sin x) + K