What is the indefinite integral of $sin (lnx) dx$ ?

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Answer 1 · 2016-07-05T09:58:58

$x/2 (Sin(Log_e(x))-Cos(Log_e(x)))$

$e^{i log_e x} = cos(log_e x)+i sin(log_e x) = e^{log_e x^i}$

then

$cos(log_e x)+i sin(log_e x) = x^i$

but

$int x^i dx = 1/2(1-i)x^{1+i} = x/2(1-i)x^i$

Taking the imaginary component

$int sin(log_e x)dx = x/2 (Sin(Log_e(x))-Cos(Log_e(x)))$

What is the indefinite integral of sin (lnx) dxsin (lnx) dx?