Integrate the following functions w.r.t x: 1/(2 + 3 tan x)

Question

Integrate the following functions w.r.t x:

\(\frac{1}{2 + 3\,tan\,x}\)

1/(2 + 3 tan x)

Answer 1

Let I = ∫ 1/(2 + 3 tan x) dx

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

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

= A (2 cos x + 3 sin x) + B (-2 sin x + 3 cos x) 

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

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

2A + 3B = 1 …… (1) 

and 3A – 2B = 0 ……. (2)

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

4A + 6B = 2 

9A – 6B = 0

On adding, we get