Fundamental Theorem of Calculus states that the derivative of an INDEFINITE integral gives you the original function.
But, this is a definite integral.

Be sure to multiply by the derivative of the bounds.

$2x\cos\left(e^{x^2}\right)−\left(0\right)\cos\left(e^0\right)=2x\cos\left(e^{x^2}\right)$

Refer to the hints in part (a). Or, notice that the definite integral equals a constant. What is the derivative of a constant?

$\text{Translation: }\int_{1}^{10}\frac{d}{dx}\left(x\log(x)\left [ \text{cos}\left ( \pi e^{x^{2}-100} \right ) \right ]\right)dx$