What is Integration By Parts? A 5 Minute Introduction

[Greg Bernhardt]

Definition/Summary
In this article, we shall learn a method for integrating the product of two functions. This method is derived from the ‘product rule’ for differentiation, but can only be applied to integrate products of certain types.
Equations
\int u dv=uv – \int v du
where u and v are functions of one variable; x, say.
Extended explanation
As, you can see in the equation, it contains two variable, namely ‘u’ and ‘v’. These variables are actually the representation of two functions and thus, the above rule can also be stated as:
\int f(x) \ g(x) \ dx=~ f(x)\int g(x) \ dx \ -~\int \left[ \ f'(x) \int g(x) \ dx \ \right] \ dx
The most important step of initiating such problems would be the determination of u and v from the given function. This can be done by using the following order:
L- Logarithmic
I- Inverse trigonometric
A- Algebraic
T- Trigonometric
E- Exponential
(Or can be remembered as ‘LIATE”)
Thus, out of the two given function, whichever comes...

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[fresh_42]

Integration by parts is the product or Leibniz rule of differentiation: $\int (fg)' - \int f\,'g = int fg'$.
Cp.: https://www.physicsforums.com/threads/what-is-the-product-rule.762916/#post-6161880

 

[wrobel]

perhaps, this will also be of some use https://www.physicsforums.com/threads/on-integration-by-parts.873148/