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Using linearity and bounds-reversal to evaluate a combined integral

Suppose ∫15f(x) dx=8\int_1^5 f(x)\,dx = 8 and ∫15g(x) dx=3\int_1^5 g(x)\,dx = 3.

Use the given information to evaluate ∫51[4f(x)−2g(x)]dx.\int_5^1 \left[4f(x) - 2g(x)\right]dx.

Hint

First apply linearity to split the integral over [1,5][1,5] into pieces involving ff and gg separately, then use the property that reversing the limits of integration negates the integral.

Step-by-step solution

By linearity, ∫15[4f(x)−2g(x)] dx=4∫15f(x) dx−2∫15g(x) dx=4(8)−2(3)=32−6=26\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 55 to 11 (bounds reversed), we use ∫51h(x) dx=−∫15h(x) dx\int_5^1 h(x)\,dx = -\int_1^5 h(x)\,dx. Applying this to our result gives ∫51[4f(x)−2g(x)] dx=−26\int_5^1 [4f(x)-2g(x)]\,dx = -26.

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