Question

if \\(\int_{4}^{-10} g(x) dx = -3\\) and \\(\int_{4}^{6} g(x) dx = 5\\), then \\(\int_{-10}^{6} g(x) dx =\\) \\(\boldsymbol{a}\\) \\(-8\\) \\(\boldsymbol{b}\\) \\(-2\\) \\(\boldsymbol{c}\\) \\(2\\) \\(\boldsymbol{d}\\) \\(8\\)

Explanation:

Step1: Reverse integral limits sign

$\int_{4}^{-10} g(x)dx = -\int_{-10}^{4} g(x)dx = -3 \implies \int_{-10}^{4} g(x)dx = 3$

Step2: Split target integral

$\int_{-10}^{6} g(x)dx = \int_{-10}^{4} g(x)dx + \int_{4}^{6} g(x)dx$

Step3: Substitute known values

$\int_{-10}^{6} g(x)dx = 3 + 5$

Answer:

D. 8