Question

Someone help me integrate this one:

Integrate ( 5cos x +6)/( 2cos x +sin x+3) w.r.t x.

Answer 1

\[\int\limits_{}^{}\frac{5cosx+6}{2cosx+sinx+3}dx=\int\limits\frac{2(2cosx+sinx+3)+(-2sinx+cosx)}{2cosx+sinx+3}dx\]\[=\int\limits \frac{2(2cosx+sinx+3)}{2cosx+sinx+3}dx+\int\limits\frac{-2sinx+cosx}{2cosx+sinx+3}dx\]\[=\int\limits 2dx+\int\limits\frac{dy}{y} \] where y= 2cosx+sinx+3
\[=2x+\ln|y| +c\]\[=2x+\ln|2cosx+sinx+3|+c\]
please check it...