Question Detail

This integral can be solved using a substitution method. Notice that the derivative of cot(u) is -cosec2(u).Let u = cot(5x).Now, differentiate u with respect to x to find du:du/dx = d/dx(cot(5x)) = -cosec2(5x) * d/dx(5x) = -5cosec2(5x).Rearrange to express cosec2(5x) dx in terms of du:cosec2(5x) dx = (-1/5) du.Substitute u and (-1/5) du into the original integral:∫ √u * (-1/5) du= (-1/5) ∫ u1/2 duNow, integrate u1/2:= (-1/5) * (u(1/2)+1 / ((1/2)+1)) + C= (-1/5) * (u3/2 / (3/2)) + C= (-1/5) * (2/3) * u3/2 + C= (-2/15) * u3/2 + CFinally, substitute u = cot(5x) back into the expression:= (-2/15) * (cot(5x))3/2 + CThis can also be written as (-2/15)√cot3(5x) + C.This result exactly matches option D.