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I've been told to evaluate the indefinite integral of this function:

$$\int \sin {\ln {x}} dx$$

I'm supposed to make a $u$-substitution in the beginning, then complete it using integration by parts. Every time I try, I just end up going in circles. Could someone please help me? Thanks!

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1 Answer

Hint: Let $u = \ln{x}$. Then $du = \frac{1}{x} dx$, or $x du = dx$. Using the fact that $x = e^u$, we can write

$$\int \sin \ln{x} dx = \int e^u \sin{u}du$$

This is a common integral that can be done by using parts twice, or by recognizing that

$$\sin{u} = \operatorname{Im} e^{iu}$$

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