Answer
see below answer
Work Step by Step
a)
$x$ = $2tanθ$
$dx$ = $2sec^{2}θdθ$
$I$ = $\int{\frac{dx}{(x^{2}+4)^{2}}}$
$I$ = $\int{\frac{2sec^{2}θdθ}{(4tan^{2}θ+4)^{2}}}$
$I$ = $\frac{1}{8}\int{\frac{sec^{2}θdθ}{(sec^{2}θ)^{2}}}$
$I$ = $\frac{1}{8}\int{cos^{2}θ}dθ$
b)
$I$ = $\frac{1}{8}\int{cos^{2}θ}dθ$
$I$ = $\frac{1}{8}\int{\frac{1+cos2θ}{2}}dθ$
$I$ = $\frac{1}{16}\int{dθ}+\frac{1}{8}\int{cos2θ}dθ$
$I$ = $\frac{1}{16}θ+\frac{1}{16}sin2θ+C$
$I$ = $\frac{1}{16}θ+\frac{1}{16}sinθcosθ+C$
c)
$x$ = $2tanθ$
$tanθ$ = $\frac{x}{2}$
$sinθ$ =$\frac{x}{\sqrt {x^{2}+4}}$
$cosθ$ =$\frac{2}{\sqrt {x^{2}+4}}$
d)
$I$ = $\frac{1}{16}θ+\frac{1}{16}sinθcosθ+C$
$I$ = $\frac{1}{16}tan^{-1}(\frac{x}{2})+\frac{x}{8(x^{2}+4)}+C$
