LPET

If the integrand f (x) is a product of polynomial, trigonometric function, exponential or logarithmic, then integration by parts may work.

Key concept:

The challenge is to choose u and dv. Choose u and dv in such a way that:

u is easy to differentiate

dv easy to integrate

Remember this term LPET
L - ln
P - polynomial
E - exponential
T - trigonometric

How to use LPET?

Choose u for the first term of LPET and dv for the second term of LPET.

Example:

f (x) is a product of polynomial and trigonometric. 
From LPET,  we have PE. 
Then, u = polynomial and dv = trigonometric

More examples:

Second example, f (x) is a product of exponential and trigonometric. 

From LPET,  we have ET. 
Then, u = exponential and dv = trigonometric


Third example, f (x) is a product of polynomial and ln. 

From LPET,  we have LP. 
Then, u = ln and dv = polynomial