Question

If $g\left(1\right)=g\left(2\right)$ then the value of ${\int }_1^2{\left[f\right\{g\left(x\right)\left\}\right]}^{-1}{f}^{\prime }\left\{g\right(x\left)\right\}{g}^{\prime }\left(x\right)dxis$

• A. $1$
• B. $2$
• C. $0$
• D. none of these

Answer 1

The correct option is C $0$

$\because g\left(1\right)=g\left(2\right){\int }_1^2{\left[f\right\{g\left(x\right)\left\}\right]}^{-1}{f}^{{}^{\prime }}\left\{g\right(x\left)\right\}{g}^{{}^{\prime }}\left(x\right)dx{\int }_1^2\frac{1}{f\left\{g\right(x\left)\right\}}{f}^{{}^{\prime }}\left\{g\right(x\left)\right\}{g}^{{}^{\prime }}\left(x\right)dx[\because \int \frac{1}{x}dx=\log x]\therefore {=\left[\log f\left(g\right(x\left)\right)\right]}_1^2\Rightarrow \log f\left(g\right(2\left)\right)-\log f\left(g\right(1\left)\right)\Rightarrow \log f\left(t\right)-\log f\left(t\right)\left[letg\right(1)=g(2)=t]=0$