Integral of a real function multiplied by an imaginary function.

[cnelson]

So I'm reviewing some mathematics for quantum mechanics and this came equation came up
\int_{-\infty}^{\infty} a \left( k \right)^{*} i \dfrac{d\,a\left(k\right)}{dk}dk.

If a \left( k \right) is constrained to be real then this integral is zero or so the text says. Why is this the case? Is it because this is the summation of two orthogonal functions so the integral must be zero. If so how what would be the first steps to proving this to myself?

 

[mathman]

Does the text say the integral is real? The integral is i{a²(∞) -a²(-∞)}/2

I don't know anything about a, but its values at the end points may be 0.

 

[cnelson]

The text just says that a \left(k\right) is real. So its derivative is also real. But the derivative is multiplied by i making it purely imaginary. So some how when these are multiplied and then integrated the integral is zero which I don't understand. Let me know if more clarification is needed.