Let be a function continuous on the positive real axis. Assume that for all positive , the integral
depends only on (hence, is independent of our choice of ). Compute the value of
if we are given that .
First, since the integral is independent of our choice of , we may let to obtain,
Next, we can use the fact that the integral is additive with respect to the interval of integration to write,
Substituting in our expression for we then have
Now, holding fixed and differentiating both sides of this equation with respect to , we find,
by the fundamental theorem of calculus. Then, since we are given that we have,
Furthermore, since , letting gives us
You don’t need the intermediate integration step when going from A’ to f.
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