Let J(m,n) = ₀^ x^ m-1 (1+x)^ m+n dx for m,n > 0 . The value of J(8, 12) + J(9, 11) is equal to

Step-by-step solution

We are given J(m,n) = ₀^ x^ m-1 (1+x)^ m+n dx . Substitute the given values into the definition: J(8, 12) = ₀^ x^7 (1+x)²⁰ dx J(9, 11) = ₀^ x^8 (1+x)²⁰ dx Adding the two integrals, we get: J(8, 12) + J(9, 11) = ₀^ x^7 + x^8 (1+x)²⁰ dx Factor out x^7 in the numerator: J(8, 12) + J(9, 11) = ₀^ x^7(1+x) (1+x)²⁰ dx Cancel the common factor (1+x) : J(8, 12) + J(9, 11) = ₀^ x^7 (1+x)¹⁹ dx Now, match this result with the definition of J(a,b) : The numerator is x^ a-1 = x^7 a = 8 The denominator is (1+x)^ a+b = (1+x)¹⁹ a+b