Let α>0. If ∫₀^α x/(√(x+α)-√x) d x=(16+20 √2)/15, then α is equal to :

Solution

Maths sir · 4 steps

Rationalise the denominator first; the α that appears there makes the integral easy.

1. Rationalise. Multiply the top and bottom by √(x+α)+√x. The denominator becomes (x+α)-x = α, so the integrand is 1/α(x√(x+α) + x^(3/2)).
2. Easy part. ∫₀^α x^(3/2) dx = 2/5α^(5/2) = 6/15α^(5/2).
3. Substitute. In ∫₀^α x√(x+α) dx, put t = x+α, so t runs from α to 2α and the integrand is t^(3/2) - α t^(1/2). Its antiderivative is 2/5t^(5/2) - 2α/3t^(3/2). This is 4√2/15α^(5/2) at t=2α and -4/15α^(5/2) at t=α, so the integral is (4√2+4)/15α^(5/2).

   1/α·(10+4√2)/15 α^(5/2) = (16+20√2)/15 ⇒ α^(3/2)(10+4√2) = 16+20√2

4. So. Try α = 2. Then α^(3/2) = 2√2, and 2√2(10+4√2) = 20√2+16, which matches. The left side increases with α, so this is the only solution. So α = 2, option (b).

The idea to keep. When the denominator is a difference of square roots, rationalise it. Here it collapses to a constant, and what is left is a sum of power functions.

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