Integration by Parts

Integration by parts is a method of evaluating easily the integral of a product of any two functions. If u and v be any two functions of x, where us is differentiable and v is integrable then,

∫u dv/dx = dx = uv – ∫v du/dx = dx

Or ∫udv = uv – ∫v du

The above model of integration by parts is derived as under:

From the differential calculus we have,

d/dx (uv) = u (dv/dx) + v (du/dx)

= u (dv/dx = (duv/dx) – (vdu/dx)

Integrating both the sides w.r.t. x we have,

∫(udvdx/dx) = ∫(d/dx) (uv) dx – ∫v (udvdx/dx)

= ∫(udv) = ∫(duv) – ∫(vdu)

∴ ∫(udv) = uv – ∫udv

Another formidable model of integration by parts may be formulated for evaluating readily the integral of a product of any two function as under:

∫(udv) = u ∫(vdx) – ∫(du/dx) ∫(vdx) dx

The above model reads as thus,

Integral of the product = [1^st function × integral of 2^nd function] of any two functions – [Integral of (derivative of the 1^st × Integral of the 2^nd function]