If $f(x)$ is a function satisfying $f^{\prime}(x)=f(x)$ with $f(0)=1$ and $g(x)$ is a function that satisfies $f(x)+g(x)=x^{2}$. Then the value of the integral $\int_{0}^{1}f(x)g(x)dx$ is
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Logic Tip: The integral $\int x^n e^x dx$ follows a simple pattern: $e^x [x^n - nx^{n-1} + n(n-1)x^{n-2} - .......]$. For $n=2$, it's simply $e^x(x^2 - 2x + 2)$. Memorizing this pattern drastically speeds up integration by parts.