I want to compute the integral, I have borders for $x$, however I would like a function of y as the result: $$ \int_a^b \quad \frac{\partial f(y,x)}{\partial y} \exp(f(y,x)) \quad dx $$
$f(y,x)$ is a Tensor Product B-Spline.Hence: $$ f(y,x) = \sum_a \sum_b B_a(y) B_b(x) \alpha_{a,b} $$ The Derivatives and Integrals of the Basis $B$ exist, see Wikipedia. See also the basis definition below: $$ B_{i,0}(t) := \begin{cases} 1 & \text{if } t_i \leq t < t_{i+1}, \\ 0 & \text{otherwise}. \end{cases} $$ $$ B_{i,p}(t) := \dfrac{t - t_i}{t_{i+p} - t_i} B_{i,p-1}(t) + \dfrac{t_{i+p+1} - t}{t_{i+p+1} - t_{i+1}} B_{i+1,p-1}(t) $$ $$ \sum_i \alpha_i B_{i,k} = \sum_{i=r-k+2}^{s-1} k\frac{\alpha_i - \alpha_{i-1}}{t_{i+k} - t_i}B_{i,k-1} \quad \text{on} \quad [t_r, t_s] $$
I wonder if there is not a trick (instead of using the product rule) since we have $f(y,x)$ both in $u$ and $v$.
Thanks in advance for any tip!