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How can I solve the following integral? $$\int\ln(e^x\sin^3x)\,dx$$

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closed as off-topic by José Carlos Santos, Delta-u, Adrian Keister, Thomas Shelby, The Count Jun 21 at 18:47

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    $\begingroup$ Hint :$\int \ln(e^x \cdot sin^3 x) dx=\int \ln e^x dx+\int \ln(\sin^3 x)dx=\int x dx +3\int \ln(\sin x)$ $\endgroup$ – Alexdanut Jun 21 at 10:52
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    $\begingroup$ There is no closed-form antiderivative using the ordinary functions. $\endgroup$ – Yves Daoust Jun 21 at 12:33
  • $\begingroup$ @YvesDaoust is painfully right $\endgroup$ – let's have a breakdown Jun 21 at 12:42
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HINT:

$$ \begin{align} &\hphantom{=.}\int \ln(e^x\sin^3x)dx \\ &= \int \ln(e^x)+\ln(\sin^3x)dx \\ &= \int (x+3\ln(\sin x))dx \\ &= \frac{x^2}{2}+3\int \ln(\sin x)dx \end{align} $$

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    $\begingroup$ use \ before sin, cos, log and etc. to make them appear neater. $\endgroup$ – DonAntonio Jun 21 at 10:57
  • $\begingroup$ thank you, I did not know that! $\endgroup$ – Pratik Apshinge Jun 21 at 10:59
  • $\begingroup$ @Pr You're wlecome. There are severla functions that work that way: \tan x, \sqrt x = $\;\sqrt x\;$ and etc. $\endgroup$ – DonAntonio Jun 21 at 11:00
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    $\begingroup$ Unfortunately, this hint will not lead to a solution. $\endgroup$ – Yves Daoust Jun 21 at 12:32
  • $\begingroup$ @ChaseRyanTaylor: this question focuses on a definite integral. $\endgroup$ – Yves Daoust Jun 21 at 12:34

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