Evaluate the following integrals:


Evaluate the following integrals:

$\int \frac{1}{x^{3}} \sin (\log x) d x$


Let $I=\int \frac{1}{x^{3}} \sin (\log x) d x$

let $\log \mathrm{x}=\mathrm{t} \Rightarrow \frac{1}{\mathrm{x}} \mathrm{dx}=\mathrm{dt} \Rightarrow \mathrm{dx}=\mathrm{e}^{\mathrm{x}} \mathrm{dt}$

We know that

$\int e^{2 x} \sin b x d x=\frac{e^{a x}}{a^{2}+b^{2}}\{a \sin b x-b \cos b x\}+c$

$\int e^{-2 t} \sin t d t=\frac{e^{-2 t}}{5}\{-2 \sin t-\cos t\}+c$

$I=\frac{x^{-2}}{5}\{-2 \sin (\log x)-\cos (\log x)\}+c$

$=\frac{-1}{5 x^{2}}\{2 \sin (\log x)+\cos (\log x)\}+c$

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