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Evaluating a combination using linearity

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

Using the given information, evaluate ∫15[4f(x)−2g(x)]dx.\int_1^5 \left[4f(x) - 2g(x)\right]dx.

Hint

Split the integral of a sum/difference into separate integrals, then pull each constant factor out front using linearity.

Step-by-step solution

By linearity of the definite integral, ∫15[4f(x)−2g(x)] dx=4∫15f(x) dx−2∫15g(x) dx\int_1^5 [4f(x) - 2g(x)]\,dx = 4\int_1^5 f(x)\,dx - 2\int_1^5 g(x)\,dx. This lets us evaluate the combined integral using only the two known values.

Substituting the given values: 4(8)−2(3)=32−6=264(8) - 2(3) = 32 - 6 = 26. So the integral equals 2626.

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