Let ∫ (2-tan x)/(3+tan x) dx=1/2(α x+logₑ|β sin x+γ cos x|)+C, where C is the constant of integration. Then α+γ/β is equal to :

Solution

Maths sir · 5 steps

Turn tan into sin over cos, then split the numerator into a multiple of the denominator plus its derivative.

1. Clear the tan. Multiply top and bottom by cos x. The integrand becomes (2 cos x-sin x)/(3 cos x+sin x).
2. Split the numerator. The denominator is D = 3 cos x+sin x and its derivative is D' = cos x-3 sin x. Write the numerator as A D + B D'.

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

3. Match coefficients. From cos x: 3A+B=2. From sin x: A-3B=-1. Solving gives A=1/2 and B=1/2.
4. Integrate. The A part integrates to 1/2x and the B part to 1/2 logₑ|D|. So the integral is 1/2(x+logₑ|sin x+3 cos x|)+C.
5. So. α=1 and γ/β=3/1=3. Scaling β and γ together only adds a constant to C, so the ratio is fixed. Hence α+γ/β=1+3= 4, option (d).

The idea to keep. For ∫ (a sin x+b cos x)/(c sin x+d cos x) dx, write the numerator as A(denominator) + B(derivative of denominator). The answer is always Ax+B log|denominator|.

Answer from the NTA final answer key. Solution by PrepWiser.