009B Sample Midterm 3, Problem 4

Evaluate the integral:

$\int \sin(\ln x)~dx.$

Foundations:
1. Integration by parts tells us that $\int u~dv=uv-\int vdu.$
2. How could we break up $\sin(\ln x)~dx$ into $u$ and $dv?$
Notice that $\sin(\ln x)$ is one term. So, we need to let $u=\sin(\ln x)$ and $dv=dx.$

Solution:

Step 1:
We proceed using integration by parts.
Let $u=\sin(\ln x)$ and $dv=dx.$ Then, $du=\cos(\ln x){\frac {1}{x}}dx$ and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle v=x.}
Therefore, we get
$\int \sin(\ln x)~dx=x\sin(\ln x)-\int \cos(\ln x)~dx.$

Step 2:
Now, we need to use integration by parts again.
Let Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle u=\cos(\ln x)} and $dv=dx.$ Then, Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle du=-\sin(\ln x){\frac {1}{x}}dx} and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle v=x.}
Therfore, we get
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\begin{array}{rcl}\displaystyle {\int \sin(\ln x)~dx}&=&\displaystyle {x\sin(\ln x)-{\bigg (}x\cos(\ln x)+\int \sin(\ln x)~dx{\bigg )}}\\&&\\&=&\displaystyle {x\sin(\ln x)-x\cos(\ln x)-\int \sin(\ln x)~dx.}\\\end{array}}}

Step 3:
Notice that the integral on the right of the last equation is the same integral that we had at the beginning.
So, if we add the integral on the right to the other side of the equation, we get
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 2\int \sin(\ln x)~dx=x\sin(\ln x)-x\cos(\ln x).}
Now, we divide both sides by 2 to get
$\int \sin(\ln x)~dx={\frac {x\sin(\ln x)}{2}}-{\frac {x\cos(\ln x)}{2}}.$
Thus, the final answer is
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \int \sin(\ln x)~dx={\frac {x}{2}}(\sin(\ln x)-\cos(\ln x))+C.}

Final Answer:
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\frac {x}{2}}(\sin(\ln x)-\cos(\ln x))+C}

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