Question

Write the coefficient a, b, c of which the value of the integral $\underset{-3}{\overset{3}{\int }}\left(a{x}^2+bx+c\right)dx$ is independent.

Answer 1

$$\begin{gathered} {\int }_{-3}^3\left(a{x}^2+bx+c\right)dx \\ ={\left[a\frac{x^3}{3}+b\frac{x^2}{2}+cx\right]}_{-3}^3 \\ =9a+\frac{9}{2}b+3c+9a-\frac{9}{2}b+3c \\ =18a+6c \end{gathered}$$

Hence, the given integral is independent of b