Question

Solve this definite integral :

[ hint : just do some changes according to the options ]​

Answer 1

Answer:

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Answer 2

Answer:

Answer seems to be the option  3) .. i mean C)

1/2 * Log [Tan (pi/12 + x/2) ]  + K  ,  K = integration constant.

Step-by-step explanation:

Given the denominator in the integrand function

  = Dr =  Cos x +  √3 Sin x

  = 2 * [ 1/2 * Cos x +  √3/2 * SIn x ]

  = 2 * { COs π/3 * Cos x + Sin π/3 * Sin x ]

  = 2 * Cos (x - π/3)

So the integrand inside the integral is  =  1/2 * Sec (x - π/3)

Let y =  x - π/3.

    dy = dx

So the integral becomes simply

 = Integral   1/2 * Sec y  * dy

 = 1/2 * Log [Sec y + Tan y ] + K      using a formula.

  = 1/2 * Log [ Sec (x -π/3) + Tan (x - π/3)  + K

Using another formula :    Integral Sec y dy = Log {Tan (π/4 + y/2)  ] + K

So  answer =  1/2 * Log [ Tan { π/4 + (x - π/3)/2 } ] + K

       = 1/2 Log [ Tan { π/12 + x/2)  ] + K

So the option c) is the correct option.