Using linearity and bounds-reversal to evaluate a combined integral

First reverse the order of the bounds: $\int_5^1 \left[2f(x)-3g(x)\right]dx = -\int_1^5 \left[2f(x)-3g(x)\right]dx$, since swapping the limits of a definite integral negates its value.

Now apply linearity to split the integral: $\int_1^5 \left[2f(x)-3g(x)\right]dx = 2\int_1^5 f(x)\,dx - 3\int_1^5 g(x)\,dx = 2(8) - 3(3) = 7$. Therefore the original integral is $-7$.