Question #668e4

1 Answer
Dec 13, 2017

int ((2sinx+cosx)*dx)/(7sinx-5cosx)=9/74*x+17/74Ln(7sinx-5cosx)+C

Explanation:

I used 2sinx+cosx=a*(7sinx-5cosx)+b*(7cosx+5sinx) equation for calculating this integral,

(7a+5b)*sinx+(7b-5a)*cosx=2sinx+cosx

After equating coefficients, 7a+5b=2 and 7b-5a=1

After solving these equations, I found a=9/74 and b=17/74

Thus,

int ((2sinx+cosx)*dx)/(7sinx-5cosx)

=9/74*int ((7sinx-5cosx)*dx)/(7sinx-5cosx)+17/74*int ((7cosx+5sinx)*dx)/(7sinx-5cosx)

=9/74*int dx+17/74*int ((7cosx+5sinx)*dx)/(7sinx-5cosx)

=9/74*x+17/74Ln(7sinx-5cosx)+C