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

First, use linearity if the integral to write

$${\begin{aligned}\int _{0}^{3}(f(x)g(x))dx=\int _{0}^{3}f(x)dx+\int _{0}^{3}g(x)dx.\end{aligned}}$$

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

$${\begin{aligned}\int _{0}^{3}(f(x)g(x))dx=\int _{0}^{5}f(x)dx-\int _{3}^{5}f(x)dx+\int _{0}^{3}g(x)dx=3-2+7=8.\end{aligned}}$$