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Integration by Parts
Integration by parts is a mathematical technique that allows one to relate the integral of a product of functions to the integral of their derivative and antiderivative. It is a basic restatement of the product rule of differentiation applied to integration.
Integration of a product of two functions
Given the product of functions f(t) and g(t), its integral is:
(1) t2⌠⌡t1d(f(t)g(t))dt  = f(t)g(t)|t2t1     t2⌠⌡t1f(t)d(g(t))dt + t2⌠⌡t1g(t)d(f(t))dt  = f(t)g(t)|t2t1
Here, the the fundamental theorem of calculus is applied. Therefore, in order to compute any one term in the integral one can use the following relation:
(2) t2⌠⌡t1f(t)d(g(t))dt = f(t)g(t)|t2t1 − t2⌠⌡t1g(t)d(f(t))dt
Integration of a product of multiple functions
In general, the theorem may be applied for a product of n factors:
(3) t2⌠⌡t1d(n∏i = 1fi(t))dt  = n∏i = 1fi(t)|t2t1     n⎲⎳j = 1t2⌠⌡t1n∏i = 1(i ≠ j)fi(t)d(fj(t))  = n∏i = 1fi(t)|t2t1
Therefore, in order to compute any one of the n terms on the left hand side, one can rearrange the above expression.