Divide [0,1] into N subintervals $\displaystyle (x_k,x_{k+1}), k=0,...,N-1$ using $\displaystyle x_k=kh,k=0,...,N, h=\frac{1}{N}, N\geq 2$.

Let $\displaystyle \phi_k(x)=\left(1-\left| \frac{x-x_k}{h}\right|\right)_{+}, k=1,...,N-1$ and let p,q be constant functions over [0,1].

I'm trying to find the piecewise solution to the integral $\displaystyle \int_0^1\left[ p \phi_j' \phi_i' + q \phi_j \phi_i\right]\ dx$. The solution is given in my notes without any explanation so I must be missing something because I'm not sure how to do it.

If someone can show me how to find e.g. $\displaystyle \int_0^1 \phi_j \phi_i \ dx$ then I should be able to do the rest. I can see that the solution will only be non-zero in the region |i-j|=1 and i=j but I'm struggling from here.