The function f(x) = ₀^x dt 1 + t satisfies which of the following differential equations?

Step-by-step solution

Given f(x) = ₀^x dt 1 + t Using the half-angle formula 1 + t = 2 ^2(t/2) , we get: f(x) = ₀^x dt 2 ^2(t/2) = 1 2 ₀^x ^2(t/2) dt Integrating, we obtain: f(x) = [ (t/2) ]₀^x = (x/2) Differentiating f(x) with respect to x : df dx = 1 2 ^2(x/2) Using the identity ^2(x/2) = 1 + ^2(x/2) : df dx = 1 2 ( 1 + ^2(x/2) ) Substituting f(x) = (x/2) : df dx = 1 2 ( 1 + [f(x)]^2 ) 2 df dx = 1 + [f(x)]^2