Integration by  parts can be regarded as the inverse to the product rule for differentiation. Suppose U(X) and V(x) are  two differentiable functions. According to the product rule

dU(x)V(x)/dx = U(x) dV(x)/dx + V(x)dU(x)/dx = U(x) dV(x)/dx+ V(x)dU(x)/dx

Integrating both sides of this equation and transposing terms, we obtain
∫U(x)dV(x)/dx dx = U(x)V(x) – ∫ V(x)dU/dx dx

This is the general formula for integration by parts.

In each application we break up the integrand into a product of two pieces U and V’:  where  V’ is easier to integrate.

With this method one can differentiate for example lnx.

Let U=lnx then dU/dx = 1/x and dU = dx/x.

and dV = dx → V=x.

∫lnxdx = x lnx – ∫x1/xdx = xlnx – x + C

Q.E.D.