A) \[x\,{{f}^{-1}}(x)+C\]
B) \[f\left( {{g}^{-1}}(x) \right)+C\]
C) \[x{{f}^{-1}}(x)-g\left( {{f}^{-1}}(x) \right)+C\]
D) \[{{g}^{-1}}(x)+C\] [Note: Where 'C' is constant of integration.]
Correct Answer: C
Solution :
\[I=\int_{{}}^{{}}{{{f}^{-1}}}(x)dx\,put\,{{f}^{-1}}(x)=t\] \[\Rightarrow \]\[x=f(t)\]or\[dx=f'(t)dt\] \[\Rightarrow \]\[I=\int_{{}}^{{}}{t\cdot f'(t)dt=t\,f(t)-\int_{{}}^{{}}{1\cdot f(t)dt}}\] \[=t\,f\,(t)-g(t)+C\] \[={{f}^{-1}}(x)x-g\left( {{f}^{-1}}(x) \right)+C\]
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