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

Question

1366 views around the world

You can reuse this answer
Creative Commons License

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)))#