# Improper integrals

1. Feb 21, 2005

If $$\int _{-\infty} ^{\infty}f(x)\: dx$$ is convergent and $$a$$ and $$b$$ are real numbers, show that

$$\int _{-\infty} ^a f(x)\: dx + \int _a ^{\infty}f(x)\: dx = \int _{-\infty} ^b f(x)\: dx + \int _b ^{\infty}f(x)\: dx$$

I'm clueless on how to show it other than by drawing what is stated: a generic finite integral being split into two finite pieces for each arbitrary point. Is there any other way to approach this problem?

Thanks

2. Feb 21, 2005

### NateTG

Perhaps you could use something like:
$$\int _{-\infty} ^a f(x)\: dx + \int _a ^{\infty}f(x)\: dx = \int _{-\infty} ^a f(x)\: dx + \int _{a} ^b f(x)\: dx +\int _b ^{\infty}f(x)\: dx = \int _{-\infty} ^b f(x)\: dx + \int _b ^{\infty}f(x)\: dx$$

But that's really just an algebraic representation of what you're suggesting.

3. Feb 21, 2005