Let $\int f(x) dx$ be $F(x)+C$, evaluate: $$\int^t_0 f(x+T)F(t-x)dx$$
Where $T$ is a constant. Integration by parts come to mind but how do I deal with the additional terms inside the two functions?
Edit: What if we let $F(x)=1-e^{-(\frac{x}{n})^b}$, where $n$ and $b$ are additional constants.
Since $x+T\neq \pm (t-x)$, it is impossible to find a result without an integration sign in it. Let's do the calculations: $$ \begin{aligned} &\int_0^t{f}\left( x+T \right) F\left( t-x \right) dx \\ &=\int_T^{t+T}{f}\left( x+T \right) F\left( t-x \right) d\left( x+T \right) \\ &=\int_T^{t+T}{F\left( t-x \right) dF\left( x+T \right)} \\ &=F\left( t-x \right) F\left( x+T \right) \mid_{T}^{t+T}-\int_T^{t+T}{F\left( x+T \right) dF\left( t-x \right)} \\ &=F\left( t+2T \right) F\left( -T \right) -F\left( t-T \right) F\left( 2T \right) +\int_T^{t+T}{f\left( t-x \right) F\left( x+T \right) dx} \end{aligned} $$ And finally we cannot eliminate anything to simply further.
