Determine if function these functions are in $\mathcal{L}^1$

Question

I want to determine if these function are in $\mathcal{L}^1(\lambda)$:

1. $\frac{\sin(t)}{1+t^2}$

2. $\frac{\cos(t)}{1+|t|}$

3. $\frac{1}{e^{t^2}}$

4. $\frac{1}{e^{t}}$

My idea was:

1. Since $\frac{sin(t)}{1+t^2}$ is Riemann integrable then it is also in $\mathcal{L}^1$. However I am not sure how to show it is Riemann integrable. It is enough to show that $\int \frac{\sin(t)}{1+t^2} dt \leq \int |\frac{1}{t^2}|dt < \infty$ because it is a convergent p-series?

2. Here I don't know where to start. I have thought about using continuity a.e. so we know it is Riemann integrable. And from there we know it is Lebesgue integrable. But all 4 functions are continuous.

3. Here I see that $\frac{1}{e^{t^2}}\leq 1$ for all t but it does not tell if the integral is finite. Furthermore the integral is taken on the entire real line

4. This one I am lost too :(

Any hints would be appreciated

Answer 1

Note that every function with convergent improper Riemman integrals is not necesserily Lebesgue integrable.

Take for instance $\frac{\sin{x}}{x}$

So for the fist function we have that $|\frac{\sin{t}}{t^2+1}|\leq \frac{1}{1+t^2} \in L^1$

Now $$\int \frac{|\cos{t}|}{|t|+1} \geq \sum_{k=0}^{+\infty} \int_{2k\pi}^{2k\pi+\frac{\pi}{4}}\frac{|\cos{t}|}{1+|t|}dt=+\infty$$ because in those intervals $\cos{t} \geq c$ for some $c >0$ where $c$ is independent of $k$

For the third function use that $e^{t^2} \geq 1+t^2$

And for the fourth function $\int e^{-t}dt \geq \int_{-\infty}^0 e^{-t}dt=+\infty$