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education/math/MATH1220 (calc II)/Integration by Parts.md
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@ -5,6 +5,7 @@ $$ \int udv = uv - \int vdu $$
$$ \frac{d}{dx}(f(x)g(x)) = f'(x)g(x) + f(x)g'(x) $$ $$ \frac{d}{dx}(f(x)g(x)) = f'(x)g(x) + f(x)g'(x) $$
1. Integrating both sides, we get: 1. Integrating both sides, we get:
$$\int \frac{d}{dx} (f(x)g(x))dx = \int [f'(x)g(x) + f(x)]$$ $$\int \frac{d}{dx} (f(x)g(x))dx = \int [f'(x)g(x) + f(x)]$$
2. Through the distributive property of integrals,
2. Therefore: $$ = \int f'(x)g(x)dx + \int f(x)g'(x)dx $$
$$$$ 3. Therefore:
$$f(x)g(x) = \intf'(x)g(x)dx $$