Answer
\[6-3\sqrt 3\]
Work Step by Step
Let \[I=\int_{0}^{3}\frac{x}{\sqrt{36-x^2}}dx\]
\[I=\frac{-1}{2}\int_{0}^{3}\frac{-2x}{\sqrt{36-x^2}}dx\]
Put \[t=36-x^2\;\;\Rightarrow \;\; dt=-2x\:dx\]
at $x=0\;\Rightarrow \;t=36$
and
$x=3\;\Rightarrow \;t=27$
\[\Rightarrow I=\frac{-1}{2}\int_{36}^{27}t^{\frac{-1}{2}}\:dt\]
\[\Rightarrow I=-\left[\sqrt t\right]_{36}^{27}\]
\[\Rightarrow I=-\left[\sqrt {27}-\sqrt {36}\right]\]
\[\Rightarrow I=6-3\sqrt 3\]
Hence, \[\int_{0}^{3}\frac{x}{\sqrt{36-x^2}}dx=6-3\sqrt 3\]
