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:

**∫ x**^{3}**e**^{x}** dx = x**^{3}**e**^{x }**– 3x**^{2}**e**^{x}** + 6xe**^{x}** – 6e**^{x}** + C**

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

__Derivative____Integration__

x^{3} e^{x}

**1st: **3x^{2} e^{x}

**2nd: **6x e^{x}

**3rd: **6 e^{x}

**4th: **0 e^{x}

**5th: **0 e^{x}

...

**∫ x**^{3}**e**^{x}** dx =** (x^{3})e^{x }– (3x^{2})e^{x} + (6x)e^{x} – (6)e^{x} + (0)e^{x} – (0)e^{x} + (0)e^{x} – (0)e^{x} + ...

= x^{3}e^{x }– 3x^{2}e^{x} + 6xe^{x} – 6e^{x} + 0 – 0 + 0 – 0 + ..

= **x**^{3}**e**^{x }**– 3x**^{2}**e**^{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.