I have a following integral
$$\int_{0}^{T-b}f\left(\tau\right)d\tau$$
where $T-b$ is an arbitrary constant number. I try to change the limits of this integral by using an indicator function and I write
$$\int_{0}^{n}f\left(\tau\right)d\tau\boldsymbol{1}_{\tau\leq T-b}$$
where $n$ is an arbitrary value. I am not sure if it is a correct way to write the integral in this way. And also, I am trying to use the fundamental theorem of calculus as
$$f\left(n\right)\boldsymbol{1}_{n\leq T-b}$$
Am I allowed to do these operations? If not, how can I correct my mistakes?
Let $f$ be a continuous real-valued map. Letting $x=T-b$, your integral is $$ \int_{0}^{x}f(t)dt=\int_{0}^{\infty}f(t)\boldsymbol{1}_{(-\infty,x]}(t)dt. $$ If I understand your question correctly, you are interested in the derivative of the function $F:[0,\infty)\rightarrow\mathbb{R}$ defined by $$ F(y)=\int_{0}^{y}f(t)\boldsymbol{1}_{(-\infty,x]}(t)dt. $$ By the fundamental theorem of calculus, $$ F^{\prime}(y)=f(y)\boldsymbol{1}_{(-\infty,x]}(y)\qquad\text{if }y\neq x. $$ The proof of the fundamental theorem uses the mean-value theorem and as such, it requires the continuity of the integrand. Therefore, the above only holds at $y\neq x$.