Integration by Parts Miracle

Standard method of integration by parts is shown in this image (I only show (u) and (dv) for the first time I do the integration by parts).

If you have no idea of what you read you either have never taken Calc II or don’t remember it. However if you know what you are looking at you will see that we just did integration by part four times in this problem which in this problem took quite a lot of time. Now as an engineer I felt there must be an easier way to do it. One of my friends gave me a method of integration by parts that was show to him by his high school Calc teacher, observe.

If what I am doing here is not clear here is an explanation. You choose your (u) and (dv) the same as you used to do, you then derivate the left column (u) until you reach zero. Then you integrate the right column until you have the same number of rows in both columns. Then diagonally cross over the first value of (u) to the second value of (dv) then add them all together. The signs as show alternate from positive to negative for each pair, starting at positive for the first pair.

As you can see we got the same answer using both methods, but the second is much faster than the first especially if we where to have to do integration by parts more times, such as if our function would be x^10sin(x). We would have to do integration by parts 10 times, or just use the quick trick I just showed you.