in d'Alembert Solution for fixed end semi infinite string problem with wave equation $u_{tt} = c^2u_{xx}$,we get $0= \frac{f(ct)+f(-ct)}{2} + \frac{\int_{-ct}^{ct}g(s)ds}{2c}$ where $f$ and $g$ are initial displacement and initial velocity function
but then my textbook says like 'this is possible when $f$ and $g$ are odd function'.but how can i justify that,i can select two function such that this expression become $0$,can anybody please explain why they are taking like that?,what is the logic behinde this? for details http://people.uncw.edu/hermanr/pde1/dAlembert/dAlembert.htm
As you wrote, if we take $x=0$ in d'Alembert's solution we get \begin{align} u(0,t)&=\frac{1}{2}[f(ct)+f(-ct)]+\frac{1}{2c}\int_{-ct}^{ct}g(s)\,ds \\ &=\frac{1}{2}[f(ct)+f(-ct)]+\frac{1}{2c}\int_{0}^{ct}[g(s)+g(-s)]\,ds. \tag{1} \end{align} If $f$ and $g$ are odd functions, then $f(-ct)=-f(ct)$ and $g(-s)=-g(s)$, hence the RHS of $(1)$ vanishes for all $t$, i.e., the boundary condition $u(0,t)=0$ is automatically satisfied.