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

Answer 1

Emma Aldridge

#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)))#

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Answer 2

Luke Alvarez

The indefinite integral of sin(ln(x)) dx is ∫sin(ln(x)) dx = -x cos(ln(x)) + ∫cos(ln(x)) dx.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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