Math 22 Integration by Parts and Present Value

Integration by Parts

 Let $u$ and $v$ be differentiable functions of $x$.
 
 $\int udv=uv-\int vdu$

Exercises Use integration by parts to evaluation:

1) $\int \ln xdx$

Solution:
Let $u=\ln x$, $>du={\frac {1}{x}}dx$
and $dv=dx$ and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle v=x}
Then, by integration by parts: $\int \ln xdx=x\ln x-\int x{\frac {1}{x}}dx=x\ln x-\int dx=x\ln x-x+C$

2) $\int xe^{3x}dx$

Solution:
Let $u=x$, $du=dx$
and $dv=e^{3x}dx$ and $v={\frac {1}{3}}e^{3x}$
Then, by integration by parts: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int xe^{3x}dx=x\frac{1}{3}e^{3x} -\int\frac{1}{3}e^{3x} dx=x\frac{1}{3}e^{3x}-\frac{1}{9}e^{3x} }

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