# Integral involving natural log exponential and sin: $\int\ln(e^x\sin^3x)\,dx$ [closed]

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

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

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

\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}
• @Pr You're wlecome. There are severla functions that work that way: \tan x, \sqrt x = $\;\sqrt x\;$ and etc. – DonAntonio Jun 21 at 11:00