Science:Math Exam Resources/Courses/MATH101 B/April 2025/Question 01 (a)/Solution 1

First, use linearity if the integral to write

$$\begin{array}{r}\underset{0}{\overset{3}{\int }}(f(x)g(x))dx=\underset{0}{\overset{3}{\int }}f(x)dx+\underset{0}{\overset{3}{\int }}g(x)dx.\end{array}$$

Next, we write $\underset{0}{\overset{3}{\int }}f(x)dx=\underset{0}{\overset{5}{\int }}f(x)dx-\underset{3}{\overset{5}{\int }}f(x)dx$ and use the given values of the integrals to compute

$$\begin{array}{r}\underset{0}{\overset{3}{\int }}(f(x)g(x))dx=\underset{0}{\overset{5}{\int }}f(x)dx-\underset{3}{\overset{5}{\int }}f(x)dx+\underset{0}{\overset{3}{\int }}g(x)dx=3-2+7=8.\end{array}$$