Question

Which of the following procedures will compute the integral

∫((3e^2x + 3) / (e^3x − 3e^2x + 3)) dx?

Question 2 options:

A. Use the substitution u=ex and then use long division on the resulting rational function inside the integral. Now apply the method of partial fraction decomposition on one of the terms in the integral.


B. Use the substitution u=ex and then proceed with integration by partial fraction decomposition of the resulting integral.


C. Use the substitution u=e2x and then proceed with integration by partial fraction decomposition of the resulting integral.


D. Use the substitution u=ex and then proceed with inverse trigonometric substitution in the resulting integral.

Answer 1

I liked B, since it gives

(3u^2+3)/(u^3 - 3u^2 + 3)

but that cubic has no rational roots, so partial fractions would be a mess.

So what do the other ideas result in?