Find the indefinite integral. Hint: First, make the substitution ; then, integrate by parts.

Question

Find the indefinite integral.  Hint: First, make the substitution  ; then, integrate by parts.

Quizplus · Accepted Answer

Making the substitution u = ln(x), we have du = 1/x dx. The integral then becomes ∫ x^2 ln(x) dx = ∫ u^2 e^u du. Integrating by parts with u^2 as the "u" term and e^u as the "dv" term, we get:

∫ u^2 e^u du = u^2 e^u - 2 ∫ u e^u du = u^2 e^u - 2 (u e^u - ∫ e^u du) = u^2 e^u - 2 u e^u + 2 e^u + C

Substituting back in for u, we have:

∫ x^2 ln(x) dx = ∫ u^2 e^u du = (ln(x))^2 x - 2 ln(x) x + 2 x + C = 11eb12b0_a95c_146d_b394_c92e2440f93b_TB7866_11 + C

Find the Indefinite Integral