Using the substitution $x = \tan \theta$ show that $\int\limits_0^1 {\frac{1}{{{{\left( {{x^2} + 1} \right)}^2}}}{\text{d}}x = } \int\limits_0^{\frac{\pi }{4}} {{{\cos }^2}\theta {\text{d}}\theta }$.

Hence find the value of $\int\limits_0^1 {\frac{1}{{{{\left( {{x^2} + 1} \right)}^2}}}{\text{d}}x}$.

let $x = \tan \theta$

$\Rightarrow \frac{{{\text{d}}x}}{{{\text{d}}\theta }} = {\sec ^2}\theta$     (A1)

$\int {\frac{1}{{{{({x^2} + 1)}^2}}}{\text{d}}x = \int {\frac{{{{\sec }^2}\theta }}{{{{({{\tan }^2}\theta + 1)}^2}}}{\text{d}}\theta } }$     M1

Note:     The method mark is for an attempt to substitute for both $x$ and ${\text{d}}x$.

$= \int {\frac{1}{{{{\sec }^2}\theta }}{\text{d}}\theta }$ (or equivalent)     A1

when $x = 0,{\text{ }}\theta = 0$ and when $x = 1,{\text{ }}\theta = \frac{\pi }{4}$     M1

$\int\limits_0^{\frac{\pi }{4}} {{{\cos }^2}\theta {\text{d}}\theta }$    AG

[4 marks]

$\left( {\int\limits_0^1 {\frac{1}{{{{\left( {{x^2} + 1} \right)}^2}}}{\text{d}}x}  = \int\limits_0^{\frac{\pi }{4}} {{{\cos }^2}\theta {\text{d}}\theta } } \right) = \frac{1}{2}\int\limits_0^{\frac{\pi }{4}} {\left( {1 + \cos 2\theta } \right){\text{d}}\theta }$   M1

$= \frac{1}{2}\left[ {\theta + \frac{{\sin 2\theta }}{2}} \right]_0^{\frac{\pi }{4}}$     A1

$= \frac{\pi }{8} + \frac{1}{4}$     A1

[3 marks]