Use integration by parts to evaluate the integral . Note that evaluation may require integration by parts more than once.

Question

Use integration by parts to evaluate the integral  . Note that evaluation may require integration by parts more than once. ​

Quizplus · Accepted Answer

Let us use integration by parts with:

u = x^2    dv = tan(x) dx
du = 2x dx   v = -ln|cos(x)|

Then:

∫ x^2 tan(x) dx = -x^2 ln|cos(x)| - 2∫x cos(x) dx

Applying integration by parts once more with:

u = x     dv = cos(x) dx
du = dx  v = sin(x)

We get:

∫ x^2 tan(x) dx = -x^2 ln|cos(x)| - 2(x sin(x) + cos(x)) + C

Simplifying, we have:

∫ x^2 tan(x) dx = x(2 - x^2) ln|cos(x)| - 2x sin(x) - 2cos(x) + C

Therefore, the answer is D.

Use Integration by Parts to Evaluate the Integral