Integration by parts

If a function f is a product of two functions u and v, the product rule yields the following differential formula

d(uv) = udv + vdu

Therefore,

which can be rearranged to

This is the formula presented in most calculus books.

A useful example to consider is the real part of a Fourier transform. In general we have

As we have seen this can be solved for the exponential function f(t) = e^-t/t to yield two terms, one real and one imaginary. We can solve for the real and imaginary parts separately using the identity

so that

From this form it is clear that we have separated the real and imaginary parts prior to carrying out the integrals.

Returning the example of the exponential function e^-t/t the real part is

Now we apply the integration by parts.

Let u = cos(wt) and dv = e^-t/tdt. Then v = te^-t/t and du = wsin(wt)dt. Using the formula for integration by parts (and ignoring the factor of 1/p) we have

Note that the first term on the right hand side evalutes to t. We can perform the same integration by parts for the second integral on the right hand side.

Note that the first term on the right hand side evaluates to zero. Substituting the second integral into the first we find

Solving for the integral term we find

which is exactly the same result as we obtain for the real part of the Fourier transform of e^-t/t (when multiplied by the factor of 1/p).