Using linearity and bounds shifting to evaluate a combined integral

First evaluate the integral with the bounds in their given order, $1$ to $5$, using linearity: $\int_1^5 [4f(x)-2g(x)]\,dx = 4\int_1^5 f(x)\,dx - 2\int_1^5 g(x)\,dx = 4(8) - 2(3) = 32 - 6 = 26$.

Since the problem asks for the integral from $5$ to $1$ (bounds reversed), we use the property $\int_5^1 h(x)\,dx = -\int_1^5 h(x)\,dx$. Applying this gives $\int_5^1 [4f(x)-2g(x)]\,dx = -26$.