Let the domain of the function
f(x) = log ₄( log ₅( log ₃(18x - x² - 77) ) ) be (a, b). Then the value of the integral ∫ₐᵇ (sin ³x)/(sin ³x + sin ³(a + b - x) dx is equal to _____________.

Solution

Maths sir · 4 steps

Peel the logs from outside in for the interval, then the King property gives half its length.

1. Outer logs first. log₄(u) needs u > 0, so log₅(log₃(g)) > 0. That means log₃(g) > 1, so g > 3, where g = 18x - x² - 77. This one condition already makes the inner logs positive.
2. Solve for x. 18x - x² - 77 > 3 gives x² - 18x + 80 < 0, that is (x - 8)(x - 10) < 0. So 8 < x < 10, giving a = 8 and b = 10.
3. King property. Call the integral I. Replacing x by a + b - x keeps the limits and swaps the two sine terms, so I = ∫ₐᵇ sin³(a+b-x)/(sin³ x + sin³(a+b-x)) dx. Add this to the original form.

   2I = ∫ₐᵇ 1 dx = b - a = 10 - 8 = 2

4. So. I = 2/2 = 1.

The common slip: 2. Stopping at 2I = b - a gives 2. That is twice the integral, so divide by 2 at the end.

The idea to keep. When the numerator is one of two terms and the denominator is their sum, with the second term being f(a+b-x), the integral is always (b-a)/2.

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