The Taylor series of the function

$$f(x) = \int\_{1}^{\sqrt{x}} \ln(xy)+ y^{3x} dy + e^{2x}$$

at the point $x = 1$ is 

$$e^2 + (x-1)\left(2e^2+\frac{1}{2}\right) + \frac{(x-1)^2}{2}\left(4e^2+\frac{7}{4}\right)$$

which I calculated using the Leibniz rule.

* How can I estimate the remainder term of second order for f(2) ? (The second derivate is already very complicated).
* Is there a method to calculate higher derivatives of parameter integrals easier than simply applying the Leibniz rule repeatedly?