If k is a positive integer such that ₀^ 2 dx k + x = 3 3 , then the value of k is:

Step-by-step solution

Let I = ₀^ 2 dx k + x . Using the half-angle substitution, let ( x 2 ) = t . Then dx = 2dt 1+t^2 and x = 2t 1+t^2 . The limits change from x = 0 2 to t = 0 1 . Substituting these into the integral: I = ₀^1 2dt 1+t^2 k + 2t 1+t^2 = ₀^1 2dt k(1+t^2) + 2t = 2 k ₀^1 dt t^2 + 2 k t + 1 Completing the square in the denominator: I = 2 k ₀^1 dt (t + 1 k )^2 + 1 - 1 k^2 = 2 k ₀^1 dt (t + 1 k )^2 + k^2-1 k^2 Using the standard integral formula dx x^2+a^2 = 1 a ⁻¹ ( x a ) with a = k^2-1 k : I = 2 k k k^2-1 [ ⁻¹ ( t + 1 k k^2-