Question

Let $f$ be a one-to-one continuous function such that $f(2)=3$ and $f(5)=7$. Given $\int\limits_2^5 f( x ) dx =17$, then the value of the definite integral $\int\limits_3^7 f^{-1}( x ) dx$ equals

• A. 10
• B. 11
• C. 12
• D. 13

Answer 1

Answer: (C)

$y=f(x) \Rightarrow x=f^{-1}(y)$ and $d y=f^{\prime}(x) d x$ now $I=\int\limits_3^7 f^{-1}(x) d x=\int\limits_3^7 f^{-1}(y) d y=\int\limits_2^5 x f^{\prime}(x) d x ;($ when $y=3$ then $x=2$ and $y=7$ then $x=5$ ) hence $I=\int\limits_2^5 x f^{\prime}(x) d x$. Integrating by parts gives, $I =\left. x f( x )\right|_2 ^5-\int\limits_2^5 f ( x ) dx $ $I =5 \cdot 7-2 \cdot 3-17=35-6-17=12$