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