# Suppose int_5^6f(x)dx = 6 and int_5^6g(x)dx=2. Evaluate int_5^6(4f(x)-2g(x))dx. Remember to include a "+ C"if appropriate.

Suppose ${\int }_{5}^{6}f\left(x\right)dx=6\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{\int }_{5}^{6}g\left(x\right)dx=2$.
Evaluate ${\int }_{5}^{6}\left(4f\left(x\right)-2g\left(x\right)\right)dx$.
Remember to include a "+ C"if appropriate.
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Theodore Schwartz
Step 1
Use the basic properties of definite integrals to find the sum of these integrals.
Step 2
${\int }_{5}^{6}\left(4f\left(x\right)-2g\left(x\right)\right)dx={\int }_{5}^{6}4f\left(x\right)dx-{\int }_{5}^{6}2g\left(x\right)dx$
$=4{\int }_{5}^{6}f\left(x\right)dx-2{\int }_{5}^{6}g\left(x\right)dx$
$=4\left(6\right)-2\left(2\right)$
$=24-4=20$