(3) int dfrac (1)({(1+{x)^2)}^dfrac (3{2)}}dx;

步骤 1：代换
令 $x = \tan y$，则 $dx = \sec^2 y \, dy$。代入原积分，得到
$$
\int \dfrac {1}{{(1+{x}^{2})}^{\dfrac {3}{2}}}dx = \int \dfrac {\sec^2 y}{{(1+\tan^2 y)}^{\dfrac {3}{2}}}dy
$$
步骤 2：化简
由于 $1 + \tan^2 y = \sec^2 y$，则
$$
\int \dfrac {\sec^2 y}{{(1+\tan^2 y)}^{\dfrac {3}{2}}}dy = \int \dfrac {\sec^2 y}{{(\sec^2 y)}^{\dfrac {3}{2}}}dy = \int \dfrac {\sec^2 y}{\sec^3 y}dy = \int \dfrac {1}{\sec y}dy = \int \cos y \, dy
$$
步骤 3：积分
$$
\int \cos y \, dy = \sin y + C
$$
步骤 4：回代
由于 $x = \tan y$，则 $y = \arctan x$，所以
$$
\sin y = \sin(\arctan x) = \dfrac{x}{\sqrt{1+x^2}}
$$