Question

${\int }_a^{b}f\left(x\right)dx=-{\int }_{a+c}^{b+c}f(x-c)dx$.

• A. True
• B. False

Answer 1

The correct option is B False

$={\int }_{a+c}^{b+c}f(x-c)dx$

$={\int }_{b+c}^{a+c}f(x-c)dx[\because {\int }_a^{b}f(x)dx=-{\int }_b^{a}f(x\left)dx\right]$

Let $x-c=z\Rightarrow dx=dz$

$={\int }_b^{a}f\left(z\right)dz$

$={\int }_b^{a}f\left(x\right)dx[\because {\int }_a^{b}f(z)dz={\int }_a^{b}f(x\left)dx\right]$

$=-{\int }_a^{b}f\left(x\right)dx$

$\because$ The above statement is false.