If ∫₀^(π/4) (sin ² x)/(1+sin x cos x) d x=1/a log ₑ(a/3)+π/(b √3), where a, b ∈ N, then a+b is equal to _________.

Solution

Maths sir · 5 steps

Divide top and bottom by cos² x, put t = tan x, and the integral turns into a clean rational function.

1. Switch to tan. Divide numerator and denominator by cos² x to get (tan² x)/(sec² x + tan x). Put t = tan x, so dx = dt/(1+t²), sec² x = 1 + t², and the limits become 0 to 1.

   I = ∫₀¹ t²/(t²+1)(t²+t+1) dt

2. Split the fraction. Since (t²+t+1)-(t²+1) = t, we have t/(t²+1)(t²+t+1) = 1/(t²+1) - 1/(t²+t+1). Multiply both sides by t, so I = ∫₀¹ t/(t²+1) dt - ∫₀¹ t/(t²+t+1) dt.
3. First integral. ∫₀¹ t/(t²+1) dt = 1/2 ln(t²+1)|₀¹ = 1/2 ln 2.
4. Second integral. Write t = 1/2(2t+1) - 1/2. This gives 1/2 ln 3 - 1/2·2/√3(tan⁻¹√3 - tan⁻¹ 1/√3) = 1/2 ln 3 - 1/√3·π/6.
5. So. I = 1/2 ln 2 - 1/2 ln 3 + π/6√3 = 1/2 ln 2/3 + π/6√3. Matching the given form, a = 2 and b = 6, so a + b = 8.

The common slip: 5. Dropping the 1/2 from t = 1/2(2t+1) - 1/2 gives π/3√3, so b = 3 and a + b = 5.

The idea to keep. When every term is a degree-2 combination of sin x and cos x, divide by cos² x and put t = tan x. The integral becomes rational in t.

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