Question

If $g\left(x\right)$ is the inverse of $f\left(x\right)$ and $f\left(x\right)$ has domain $x\in [1,5]$, where $f\left(1\right)=2$ and $f\left(5\right)=10$ then the values of ${\int }_1^{5}f\left(x\right)dx+{\int }_2^{10}g\left(y\right)dy$

• A. $48$
• B. $64$
• C. $71$
• D. $52$

Answer 1

The correct option is A $48$

Given : $g\left(x\right)$ is inverse of $f\left(x\right)$ $x\in [1,5]$

$f\left(1\right)=2,f\left(5\right)=10$

According to question $f$ is one - one function which implies $g$ is also one - one function.

Given, $g$ is inverse of $f$.

Implies $f$ is inverse od $g$

${\int }_a^{b}f\left(x\right)dx={\int }_1^{5}f\left(x\right)dx={\left[xf\right(x)-{\int }_1^{5}x{f}^{\prime }\left(x\right)dx]}_1^5=x{g}^{-1}\left(x\right)-{\int }_1^{5}x{g}^{-1}\left(x\right)dxLetx=g\left(u\right)u=g^{-1}\left(1\right)du=g^{-1}\left(1\right)dx=x{g}^{-1}\left(x\right)-{\int }_{g^{-1}\left(1\right)}^{g^{-1}\left(5\right)}g\left(u\right)du{\int }_1^{5}f\left(x\right)dx+{\int }_2^{10}g\left(y\right)dy={\left[xf\right(x\left)\right]}_1^5-{\int }_2^{10}g\left(y\right)du+{\int }_2^{10}g\left(y\right)dy=5f\left(5\right)-1f\left(1\right)=48$

Hence the correct answer is $48$