Question

evaluate: integral from 3 to infinity: 1/(x-2)^(3/2) dx

Answer 1

\[I=\int_3^\infty\frac{1}{(x-2)^{3/2}}dx\\
u=x-2\implies du=dx\\
\text{when}\;x=3,u=1\\
\text{when}\;x=\infty,u=\infty\\
I=\int_1^\infty\frac{1}{u^{3/2}}du\\
I=\int_1^\infty u^{-3/2}du\\
I=\left[\frac{u^{-1/2}}{-1/2}\right]_1^\infty=-2\left[{1\over\sqrt{u}}\right]_1^\infty\\
I=-2\left({1\over\infty}-1\right)\\
\boxed{I=2}
\]