What would you assume at the beginning of an indirect proof for the following?
1 answer:
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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: <em>indefinite integral integration by parts reduction formula product polynomial exponential differential integral calculus</em>
_______________
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: <em>indefinite integral integration by parts reduction formula product polynomial exponential differential integral calculus</em>