Why is $\int_{-1}^1 (1-|t|) \cdot e^{-i \omega t} dt$ equal to $2 \cdot \int_0^1 (1-t) \cdot \cos{\omega t}\, dt$

Question

I have a task where I should calculate the fourier transform of

$$ \Delta(t) = \begin{cases} (1-|t|)& |t| \le 1 \\ 0 & |t| > 1 \end{cases} $$ The solution says, that $$ \int_{-1}^1 (1-|t|) \cdot e^{-i\omega t} dt = 2 \cdot \int_0^1 (1-t) \cdot \cos{\omega t}\, dt $$

I understand that the factor must be $2$ and that we only need to integrate from 0 to 1 because $\Delta(t)$ is an even function. But i don't get the last transformation.

$$ \int_{-1}^1 (1-|t|) \cdot e^{-i\omega t} dt = 2 \cdot \int_0^1 (1-t) \cdot e^{-i\omega t}\,dt = \mathop{???} = 2 \cdot \int_0^1 (1-t) \cdot \cos{\omega t}\, dt $$

Does anybody know how the transformation from $e^{-i\omega t}$ to $\cos{\omega t}$ works?

I would say that this are different functions, because normally i would say $e^{-i\omega t} = \cos{\omega t} - i \sin{\omega t}$

Answer 1

They have sort of buried the idea; $\cos(\omega t)$ is an even function of $t$ and $\sin(\omega t)$ is an odd function of $t$. If you use Euler's formula first before folding the integral in half it makes more sense.

To be more specific:

\begin{align*} \int_{-1}^{1} (1-|t|)e^{-i\omega t}dt &= \int_{-1}^{1} (1-|t|)(\cos(\omega t)-i\sin(\omega t))dt \\ &= \int_{-1}^{1} (1-|t|)\cos(\omega t)dt - i\int_{-1}^{1} (1-|t|)\sin(\omega t)dt \\ &= \int_{-1}^{1} (1-|t|)\cos(\omega t)dt +0 \end{align*}

Answer 2

You wrote

$$ \int_{-1}^1 (1-|t|) e^{-i\omega t} dt = 2\int_{0}^1 (1-t) e^{-i\omega t}dt $$

This is false, as $e^{-i\omega t}$ is not an even function. But all functions can be divided into an even part and an odd part. For $e^{-i\omega t}$, this is given by $e^{-i\omega t} = \cos(\omega t) -i\sin(\omega t)$. So we have $$ \int_{-1}^1 (1-|t|) e^{-i\omega t} dt = \int_{-1}^1 (1-|t|) \cos(\omega t) dt - i\int_{-1}^1 (1-|t|) \sin(\omega t) dt \\ = 2\int_{-1}^1 (1-t) \cos(\omega t) dt, $$ where the sine integral is zero because its integrand is odd.