Find the fourier transform of the triangle function give below.f( t)=( a+ t)/a² −a< t<0,( a-t)/a² 0< t< a,0 | t|> a please give me the solve steps by steps

To find the Fourier Transform of the given triangle function, we use the definition of the Fourier Transform:

$$F(\omega) = \int_{-\infty}^{\infty} f(t) e^{-j\omega t} dt$$

The function $f(t)$ is defined as:

$$f(t) = \begin{cases} \frac{a+t}{a^2} & -a < t < 0 \\ \frac{a-t}{a^2} & 0 < t < a \\ 0 & |t| \ge a \end{cases}$$

Due to the piecewise definition of $f(t)$, we split the integral into two parts:

$$$$

\int_{0}^{a} \frac{a-t}{a^2} e^{-j\omega t} dt

**Step 3: Evaluate the second integral $I_2 = \int_{0}^{a} \frac{a-t}{a^2} e^{-j\omega t} dt$** Let $u = a-t$, $dv = e^{-j\omega t} dt$. Then $du = -dt$, $v = \frac{e^{-j\omega t}}{-j\omega}$.

I_2 = \frac{1}{a^2} \left[ \left. (a-t) \frac{e^{-j\omega t}}{-j\omega} \right|{0}^{a} - \int{0}^{a} \frac{e^{-j\omega t}}{-j\omega} (-dt) \right]

$$I_2