Determine the most appropriate method or integral formula for evaluating the given integral. Next, evaluate the integral.
I. Integration by parts II.
III.
IV.

A) I. Integration by parts;
D)
E) not integrable by any of the techniques we have studied

Question

Determine the most appropriate method or integral formula for evaluating the given integral. Next, evaluate the integral. ​
I. Integration by parts II. 
III. 
IV.

A) I. Integration by parts; 
D) ​ 
E) not integrable by any of the techniques we have studied

Quizplus · Accepted Answer

The integral can be evaluated using integration by parts with u = ln(x) and dv = e^x dx. This leads to the formula:

∫ ln(x) e^x dx = ln(x) e^x - ∫ 1/x e^x dx

The second term can be integrated by parts with u = 1/x and dv = e^x dx, giving:

∫ ln(x) e^x dx = ln(x) e^x - (e^x/x - ∫ -e^x/x^2 dx)

Simplifying this expression gives:

∫ ln(x) e^x dx = (ln(x) - 1/x) e^x + C

Where C is the constant of integration. Therefore, the best choice is A.

Determine the Most Appropriate Method or Integral Formula for Evaluating