Question

How do you integrate `\int _ { 0} ^ { 1} \frac { x } { \sqrt { 1+ x ^ { 2} } } d x`?

Answer 1

`int_0^1x/(sqrt(1+x^2))dx=sqrt2-1`

Explanation:

Let `x=tantheta`, then `dx=sec^2theta d theta`

and `int_0^1x/(sqrt(1+x^2))dx`

= `int_0^(pi/4)tantheta/sectheta sec^2theta d theta`

= `int_0^(pi/4)tanthetasectheta d theta`

= `|sectheta|_0^(pi/4)`

= `sqrt2-1`