Mathematics

Problem

Let

f

and

g

be continuous on

[0,5]

. It is given that

\int_{0}^{5}\bigl(2f(x)+3g(x)\bigr)\,dx=46\quad\text{and}\quad\int_{0}^{5}\bigl(5f(x)-4g(x)\bigr)\,dx=23.

Using only algebraic manipulation of definite integrals, answer the following. a. (i) Determine

\displaystyle \int_{0}^{5} f(x)\,dx

. b. (ii) Hence find

\displaystyle \int_{0}^{5} g(x)\,dx

. c. (iii) Evaluate

\displaystyle \int_{5}^{0}\bigl(f(x)+g(x)\bigr)\,dx

and justify your answer.