Question

Integrate the function w..r. to x : sin (log x)

Answer 1

Let

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

Put this substitution into the equation

$\int sin(log x) dx= \int sin t.e^{t} dt$

Now integrate this by parts

$\int U.V dx= U\int V dx-\int (\frac{dU}{dx}\int Vdx) dx\\\\\int e^{t} sin\:t dt= e^{t}\int sin\:t\:dt-\int (\frac{de^{t}}{dt}\int sin tdt)dt\\\\\int e^{t}sint dt= -e^{t}cost +\int e^{t}cos t dt\\\\\int e^{t} sin t dt= -e^{t}cos t +e^{t}\int cos t dt-\int (\frac{de^{t}}{dt}\int cos t.dt) dt\\\\\int e^{t} sin t dt= -e^{t} cos t +e^{t} sin t-\int e^{t}.sint dt\\\\2\int e^{t} sin t dt= -e^{t}cost +e^{t} sin t +C\\\\\int e^{t}sin t dt= \frac{e^{t}}{2}( sin t - cos t)+C$

Undo substitution

$\int sin(log x)= \frac{e^{log x}}{2}( sin(log x)- cos(log x) )+C\\\\=\frac{ x}{2}( sin(log x)- cos(log x) )+C$