Question Detail

This integral is of the form ∫1/(a²+u²) du = (1/a)tan−¹(u/a) + C.First, rewrite the denominator to match this form: 4+9x² = 2²+(3x)².So, we have a = 2 and u = 3x.Next, find du: du = d/dx (3x) dx = 3 dx. This implies dx = du/3.Substitute these into the integral:∫(1/(2²+u²)) (du/3) = (1/3) ∫(1/(2²+u²)) du.Now apply the standard integral formula:(1/3) * (1/2)tan−¹(u/2) + C.(1/6)tan−¹(u/2) + C.Substitute back u = 3x:(1/6)tan−¹(3x/2) + C.Option C matches this result.