how many time i have to apply by parts

Question

is there a way to know how many time should I apply by parts?

as this example

x^2.5 cosh(sqrt(x))?

Answer 1

How many time i have to apply by parts. Is there a way to know how many time should I apply by parts? as this example: x^2.5 cosh(sqrt(x))?

Detailed Answer:

Here is an example that will answer your question:

∫ x³e^x dx = x³e^x – 3x²e^x + 6xe^x – 6e^x + C

We will do integration by parts using integration by parts table (here it is):

Derivative Integration

x³ e^x

1st: 3x² e^x

2nd: 6x e^x

3rd: 6 e^x

4th: 0 e^x

5th: 0 e^x

...

∫ x³e^x dx = (x³)e^x – (3x²)e^x + (6x)e^x – (6)e^x + (0)e^x – (0)e^x + (0)e^x – (0)e^x + ...

= x³e^x – 3x²e^x + 6xe^x – 6e^x + 0 – 0 + 0 – 0 + ..

= x³e^x – 3x²e^x + 6xe^x – 6e^x + C

Answer to your question: In integration by parts, eventually you run out of Derivative (turn to "0"); in this example, you run out of Derivative after the 3rd (every Derivative after that is "0" – useless). Thus, you stop at 3rd Derivative.

Answer 2

No, there is no theorem that says how many times you need to apply integration by parts. Every time it depends on the problem.