Using linearity and bounds-reversal to evaluate a combined integral

Since reversing the limits of integration negates the integral, $\int_5^1 [2f(x)-3g(x)]\,dx = -\int_1^5 [2f(x)-3g(x)]\,dx$. This lets us work with the given integrals over $[1,5]$ instead.

By linearity, $\int_1^5 [2f(x)-3g(x)]\,dx = 2\int_1^5 f(x)\,dx - 3\int_1^5 g(x)\,dx = 2(8) - 3(3) = 7$. Applying the negative sign from the bounds reversal gives $\int_5^1[2f(x)-3g(x)]\,dx = -7$.