how to calculate integrate about Heaviside everyone,here I have a question about how to calculate
 $$\int e^t H(t) dt$$
where $H(t)$ is Heaviside step function
thank you for your answering!!
Hint: The Heaviside function is defined as
\begin{aligned}
\text{H}(t) &amp; = 0, \quad t&lt;0,\\
\text{H}(t) &amp; = 1, \quad t\geq0.
\end{aligned}
That means your integrand will be zero for all $t&lt;0$, and $e^{t}$ for $t\geq1$. Can you figure out the rest?

Note: If you use the half-maximum convention though, you will find it to be:
\begin{aligned}
\text{H}(t) &amp; = 0, \quad t&lt;0,\\
\text{H}(t) &amp; = \frac{1}{2}, \quad t=0,\\
\text{H}(t) &amp; = 1, \quad t&gt;0.
\end{aligned}
If so, you will have the integrand is $\frac{1}{2}e^{t}$ in $t=0$, and the rest is similar to the above.
In fact the result is just trivially

$\int e^tH(t)~dt=e^tH(t)+C$