Question

$f\left(x\right)$ is a continuous functions for all real values of $x$ and satisfies ${\int }_n^{n+1}f\left(x\right)dx=\frac{n^2}{2},\forall nϵZ$, then ${\int }_{-3}^{5}f\left(x\right)dx$ is equal to

• A. $\frac{9}{2}$
• B. $\frac{7}{2}$
• C. $\frac{35}{2}$
• D. None of these

Answer 1

The correct option is A $\frac{9}{2}$

We have, $I={\int }_{-3}^{5}f\left(|x|\right)dx$

$I$ can also be written as

$I={\int }_{-3}^{-2}f\left(|x|\right)dx+$${\int }_{-2}^{-1}f\left(|x|\right)dx+$${\int }_{-1}^{0}f\left(|x|\right)dx+$${\int }_0^{1}f\left(x\right)dx+$${\int }_1^{2}f\left(x\right)dx+$${\int }_2^{3}f\left(x\right)dx+$${\int }_3^{4}f\left(x\right)dx+$${\int }_4^{5}f\left(x\right)dx$

$I=2{\int }_0^{1}f\left(x\right)dx+$$2{\int }_1^{2}f\left(x\right)dx+$$2{\int }_2^{3}f\left(x\right)dx+$${\int }_3^{4}f\left(x\right)dx+$${\int }_4^{5}f\left(x\right)dx$

$I=\frac{0}{2}+\frac{2}{2}+\frac{8}{2}+\frac{9}{2}+\frac{16}{2}=13+\frac{9}{2}=\frac{35}{2}$