Integrate the following functions w.r.t x: (sin x + 2 cos x)/(3 sin x + 4 cos x)

Question

Integrate the following functions w.r.t x:

\(\frac{sin\,x + 2\,cos\,x}{3\,sin\,x + 4\,cos\,x}\)

(sin x + 2 cos x)/(3 sin x + 4 cos x)

Answer 1

(sin x + 2 cos x)/(3 sin x + 4 cos x)

Put, Numerator = A (Denominator) + B [d/dx (Denominator)]

∴ sin x+ 2 cos x = A(3 sin x + 4 cos x) + B [d/dx (3 sin x + 4 cos x)]

= A(3 sin x + 4 cos x) + B (3 cos x – 4 sin x)

∴ sin x + 2 cos x = (3A – 4B) sin x + (4A + 3B) cos x

Equating the coefficients of sin x and cos x on both the sides, we get

3A – 4B = 1 …… (1) 

and 4A + 3B = 2 …… (2)

Multiplying equation (1) by 3 and equation (2) by 4, we get

9A – 12B = 3 

16A + 12B = 8

On adding, we get