Question Detail

To find y, we need to integrate the given derivative dy/dx with respect to x:y = ∫ (cos(3√x)/√x) dxWe can use a substitution method to solve this integral. Let u = 3√x.To find du, differentiate u with respect to x:du/dx = d/dx(3x1/2) = 3 * (1/2)x(1/2)-1 = (3/2)x-1/2 = 3 / (2√x).Rearrange to find (1/√x) dx in terms of du:(1/√x) dx = (2/3) du.Now, substitute u and (1/√x) dx into the integral:y = ∫ cos(u) * (2/3) duy = (2/3) ∫ cos(u) duIntegrate cos(u) with respect to u:y = (2/3) sin(u) + CFinally, substitute u = 3√x back into the expression:y = (2/3) sin(3√x) + CComparing this result with the given options, assuming the constant of integration C is zero, option D matches our derived function for y.