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education/math/MATH1220 (calc II)/Integration by Parts.md
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The integration by parts formula is:
$$ \int udv = uv - \int vdu $$
Broadly speaking, integration by parts is done by:
1. Pick a part of integral to be $u$.
2. The rest of the integral will be $dv$,
3. Compute the derivative of $u$, $du$.
4. Compute the antiderivative of $dv$
5. Substitute those values in to the integration by parts formula.
## Deriving the Integration by Parts Formula
$$ \frac{d}{dx}(f(x)g(x)) = f'(x)g(x) + f(x)g'(x) $$
1. Integrating both sides, we get:
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1. Substituting $w$ and $dw$ into the integral:
$$ \int \frac{1}{2}e^w \frac{1}{2}dw $$
2. This gives an integral that can be computed naively
$$ \int\frac{1}{2}e^{w} $$
$$ \int\frac{1}{2}e^{w}\frac{1}{2}dw = \frac{1}{4}\int e^w dw= \frac{1}{4}e^{2x} $$
4. Combining everything together, we get:
$$ \int xe^{2x}dx = (\frac{1}{2}e^{2x})(x)- (\frac{1}{4}e^2x) + C$$