Evaluate: `int(2sinx+3cosx)/(3sinx+4cosx)\ dx`

Explanation

We have given

`int(2sinx+3cosx)/(3sinx+4cosx)\ dx`

`I=int(2sinx+3cosx)/(3sinx+4cosx)\ dx`

`(2sinx+3cosx)=A(3sinx+4cosx)+B[d/(dx)(3sinx+4cosx)]`

`(2sinx+3cosx)=A(3sinx+4cosx)+B(3cosx-4sinx)`

On comparing sinx and cosx coefficients and solving them we get

`A=18/25,B=1/25`

`I=int(2sinx+3cosx)/(3sinx+4cosx)\ dx=18/25int1dx+(1/25)int(d(3sinx+4cosx))/(3sinx+4cosx)`

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