How many applications of integration by parts are required to evaluate integral (x^7)(e^x) dx?

i believe the answer is 7 because after 7 iterations, you should have gotten rid of the x^7 and be left with the integral of e^x, but i am not sure if that final integral is considered another iteration of integration by parts through

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Make it more general, and see what happens when you try to reduce the exponent of  x  in the following integral:



Now, integrate it by parts:





where 

So after one iteration, the exponent of  x  was decreased by one unit.

The question is:  after how many iterations will the exponent of  x  equals zero?

     After exactly  k  iterations, of course.

Therefore, for  k = 7, you have to apply integration by parts  7  times, to get rid of that polynomial factor. Then, there will be one last integral left to evaluate:



But this one doesn't need to be evaluated by parts. You can directly write the result:



Shortly, for the integral



you have to apply integration by parts  7  times (not  8  times).

I hope this helps. =)

Tags:  indefinite integral integration by parts reduction formula product polynomial exponential differential integral calculus

The answer is part a: use