Question

This question is designed to be answered without a calculator.
If f and g are functions such that (picture)
what is the value of (picture)
-12
-3
3
12

Answer 1

Given that

$\displaystyle \int_0^2 f(x) \, dx = 2$

$\displaystyle \int_0^2 f(x) - 2g(x) \, dx = 8$

we use the linearity of the definite integral to solve for

$\displaystyle \int_0^2 g(x) \, dx$

We have

$\displaystyle \int_0^2 f(x) - 2g(x) \, dx = \int_0^2 f(x) \, dx - \int_0^2 2g(x) \, dx$

$\displaystyle \int_0^2 f(x) - 2g(x) \, dx = \int_0^2 f(x) \, dx - 2 \int_0^2 g(x) \, dx$

Then using the given known integrals,

$\displaystyle 8 = 2 - 2 \int_0^2 g(x) \, dx \implies \int_0^2 g(x) \, dx = \boxed{-3}$