# I Complex integral of a real integrand

1. May 5, 2017

### spaghetti3451

I am trying to do the following integral:

$$\int_{\pi f}^{3\pi f} dx \sqrt{\cos(x/f)}.$$

Wolfram alpha - http://www.wolframalpha.com/input/?i=integrate+(cos(x))^(1/2)+dx+from+x=pi+to+3pi gives me

$$\int_{\pi f}^{3\pi f} dx \sqrt{\cos(x/f)} = 4f E(2) = 2.39628f + 2.39628if,$$

where E is the elliptic function.

Mathematica also gives me the same answer. How can the integral of a real integrand with real limits be complex?

2. May 5, 2017

### Orodruin

Staff Emeritus
It cannot. Your integrand is not real.

3. May 5, 2017

got it!

4. May 5, 2017

### Staff: Mentor

What's the meaning of 'f' in the limits of integration and in cos(x/f)? You don't have it in your Wolframalpha link. When I click the WA link you provided, it says that it doesn't understand the query, and gives a result of 1/2.

5. May 5, 2017

### Orodruin

Staff Emeritus
Remove the backslash ...

6. May 5, 2017

### spaghetti3451

This question is with regards to tunneling in false vacuum decay.
$f$ is a constant. Here is the link that works:

http://www.wolframalpha.com/input/?i=integrate+(cos(x))^(1/2)+dx+from+x=pi+to+3pi