Let the slope of the tangent to a curve y = f(x) at (x, y) be given by 2 tan x(cos x - y). If the curve passes through the point ( π/4,0 ), then the value of ∫₀^(π /2) y dx is equal to :

Solution

Maths sir · 4 steps

Write the slope condition as a linear equation in y; the factor sec² x does the rest.

1. Make it linear. Expanding dy/dx = 2 tan x(cos x - y) gives dy/dx + 2 tan x · y = 2 sin x, since tan x cos x = sin x. This is linear in y with P = 2 tan x and Q = 2 sin x.
2. Integrating factor. ∫ 2 tan x dx = 2 ln|sec x|, so the integrating factor is e²ˡⁿˢᵉᶜˣ = sec² x. Then y sec² x = ∫ 2 sin x sec² x dx = ∫ (2 sin x)/(cos² x) dx = 2 sec x + C.
3. Use the point. At x = π/4, y = 0, so 0 = 2√2 + C, which gives C = -2√2. Multiplying through by cos² x gives y = 2 cos x - 2√2 cos² x.

   ∫₀^(π/2) y dx = 2∫₀^(π/2)cos x dx - 2√2∫₀^(π/2)cos² x dx = 2(1) - 2√2(π/4)

4. So. Since 2√2 × π/4 = π/√2, the integral is 2 - π/√2, which is option (b).

The idea to keep. Whenever y appears only to the first power in the slope expression, rearrange to dy/dx + Py = Q and use e^(∫ P dx). Fix the constant from the given point before integrating further.

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