Integration by Parts

Integration by parts method is generally used to find the integral when the integrand is a product of two different types of functions or a single logarithmic function or a single inverse trigonometric function or a function which is not integrable directly.

If f(x) and g(x) are two differentiable functions then

d[f(x) g(x)]/dx = f′(x) g(x) + f(x) g′(x)

By definition of anti derivative

f(x) g(x) = ∫ f′(x) g(x) dx+ ∫ f(x) g′(x) dx

Rearranging

∫ f(x) g′(x) dx = f(x) g(x) − ∫ f′(x) g(x) dx

Let u = f(x) and v = g(x)

du = f′(x) dx and dv = g′(x) dx

∫ u dv = uv − ∫ v du

If integrand contains any non-integrable functions directly from the formula, like log x, tan⁻¹ x etc., take these unintegrable functions as u and other as dv.

If the integrand contains both the integrable function, and one of these is xⁿ (where n is a positive integer). Then take u = xⁿ.