Question 1. Graphical Integration Challenge: u-Substitution and Integration by Parts
Let $F(x)={\int }_{-1}^{x}f(t)\mathrm{d}t$ and $G(x)={\int }_0^{x}g(t)\mathrm{d}t$, where the graphs of $f(x)$ and $g(x)$ on $[-2,4]$ are below.
(a) Evaluate ${\int }_0^2[F(x)f(x)-G(x)g(x)]\mathrm{d}x$.
HINT: Consider a $u$-substitution (possibly multiple substitutions). You may need to invoke Part I of the Fundamental Theorem of Calculus.

Step 1 ：Let $F(x)={\int }_{-1}^{x}f(t)\mathrm{d}t$ and $G(x)={\int }_0^{x}g(t)\mathrm{d}t$, where the graphs of $f(x)$ and $g(x)$ on $[-2,4]$ are given.

Step 2 ：We are asked to evaluate ${\int }_0^2[F(x)f(x)-G(x)g(x)]\mathrm{d}x$.

Step 3 ：We can use the Fundamental Theorem of Calculus and u-substitution to simplify the integral. The Fundamental Theorem of Calculus states that if a function f is continuous over the interval [a, b] and F is an antiderivative of f on [a, b], then ${\int }_a^{b}f(x)dx=F(b)-F(a)$.

Step 4 ：For the first term, let $u=F(x)$ and $dv=f(x)dx$. Then find $du$ and $v$, and use the formula for integration by parts, $\int udv=uv-\int vdu$, to simplify the integral.

Step 5 ：For the second term, let $u=G(x)$ and $dv=g(x)dx$. Then find $du$ and $v$, and use the formula for integration by parts, $\int udv=uv-\int vdu$, to simplify the integral.

Step 6 ：However, without the actual graphs of $f(x)$ and $g(x)$, we cannot provide a numerical answer.

Step 7 ：$\boxed{{\displaystyle \text{The final answer would depend on the specific graphs of }f(x)\text{ and }g(x).}}$