partial derivative of definite intergral

Question

Question: Find the partial derivatives, $f_x(x, y)$ and $f_y(x,y)$, of the function $$f(x,y)=\int_y^xcos(3t^2+9t-1)dt$$

My attempt is as follows.

1. Substitution:

   $u=3t^2+9t-1$

   $\frac{du}{dt}=6t+9$

   $dt=\frac{1}{6t+9}du$

2. Plug in $u$:

   $\int_y^xcos(3t^2+9t-1)dt$

   $=\int_y^x\frac{cos(u)}{6t+9}du$

   and I don't know how to continue.

I've been stuck on this question for a few days now. Tried searching across all platforms but none has similar questions like this.

WolframAlpha shows an answer that involves the Fresnel C and S integrals but my class hasn't mentioned this anywhere.

Answer 1

No need to calculate the integral: $f_(x,y)=\cos(3x^{2}+9x-1)$ and $f_y((x,y)=-\cos(3y^{2}+9y-1)$. You only have to know that the derivative of an indefinite integral gives back the original function.