Question

The area bounded by the x-axis, the curve y = f(x) and the lines x = 1, x = b is equal to $\sqrt { b ^ { 2 } + 1 } - \sqrt { 2 }$ for all $b > 1$then f(x) is

• A. $\sqrt { x - 1 }$
• B. $\sqrt { x + 1 }$
• C. $\sqrt { x ^ { 2 } - 1 }$
• D. $\frac { x } { \sqrt { 1 + x ^ { 2 } } }$

Answer 1

Answer: (D)

$\frac { x } { \sqrt { 1 + x ^ { 2 } } }$

Answer 2

$\int _ { 1 } ^ { b } f ( x ) d x = \sqrt { b ^ { 2 } + 1 } - \sqrt { 2 } = \left[ \sqrt { x ^ { 2 } + 1 } \right] _ { 1 } ^ { b }$

⇒ $f ( x ) = \frac { 1 } { 2 } \cdot \frac { 2 x } { \sqrt { x ^ { 2 } + 1 } }$

Hence $f ( x ) = \frac { x } { \sqrt { 1 + x ^ { 2 } } }$.