Evaluate the following integrals : ∫(-sinx+2cosx)/(2sinx+cosx)dx ​

Question

Evaluate the following integrals :

\(\int\frac{-sin\,x+2cos\,x}{2sinx+cosx}\)dx

Answer 1

Assume,

2sinx + cosx  = t 

d(2sinx + cosx) = dt 

(2cosx - sinx)dx = dt 

Put t and dt in given equation we get

⇒ ∫\(\frac{dt}{t}\)

= In|t| + c

But,

t = 2sinx + cosx 

= ln|2sinx + cosx| + c.