Question

If $f\left(x\right)$ is monotonic differentiable function on $[a,b]$, then ${\int }_a^{b}f\left(x\right)dx+{\int }_{f\left(a\right)}^{f\left(b\right)}{f}^{-1}\left(x\right)dx$

• A. $bf\left(A\right)-af\left(B\right)$
• B. $bf\left(B\right)-af\left(A\right)$
• C. $f\left(A\right)+f\left(B\right)$
• D. $Cannotbefound$

Answer 1

The correct option is B $bf\left(B\right)-af\left(A\right)$

Consider $I={\int }_a^{b}f\left(x\right)dx+{\int }_{f\left(a\right)}^{f\left(b\right)}{f}^{-1}\left(x\right)dx$

$I_1$ $I_2$

take $I_2={\int }_{f\left(a\right)}^{f\left(b\right)}{f}^{-1}\left(x\right)dx$ put $x=f\left(t\right)$

$dx=f^1\left(t\right)dt$

$={\int }_b^a{f}^{-1}\left(f\right(+)f^{1}dt$

$={\int }_a^{b}t{f}^1\left(t\right)dt=tf\left(t\right){|}_a^b-={\int }_b^{a}f\left(t\right)dt$

$I_2+I_1=bf\left(b\right)-af\left(a\right)$

$I=bf\left(b\right)-af\left(a\right)$