Methods of Integration: Integration by Parts

If u and v are any two differentiable functions of a single variable x . Then, by the product rule of differentiation, we have 
`d/(dx)(uv)` = u`(dv)/(dx)` + v`(du)/(dx)`
Integrating both sides, we get

uv = `int u (dv)/(dx) dx + int v (du)/(dx) dx`

or `int u (dv)/(dx)dx = uv - int v (du)/(dx) dx`        ...(1)

Let u = f(x) and `(dv)/(dx)`=g(x) . Then 

`(du)/(dx) = f'(x) and v = int g(x) dx`

Therefore, expression (1) can be rewritten as
`int f(x) g(x) dx = f(x)int g(x) dx - int [ int g(x) dx] f'(x) dx`

i.e. `int f(x) g(x) dx = f(x) int g(x) dx - int [f'(x) int g(x) dx] dx`

If we take f as the first function and g as the second function, then this formula may be stated as follows: 

“The integral of the product of two functions 
= (first function) × (integral of the second function) – Integral of [(differential coefficient of the first function) × (integral of the second function)]”