From db1175540a7c6aaa276125ea16ca7fb9a6fee0f6 Mon Sep 17 00:00:00 2001 From: arc Date: Wed, 3 Sep 2025 12:35:04 -0600 Subject: [PATCH] vault backup: 2025-09-03 12:35:04 --- .../math/MATH1220 (calc II)/Integration by Parts.md | 11 +++++++++-- 1 file changed, 9 insertions(+), 2 deletions(-) diff --git a/education/math/MATH1220 (calc II)/Integration by Parts.md b/education/math/MATH1220 (calc II)/Integration by Parts.md index 477fe87..ed7bd2a 100644 --- a/education/math/MATH1220 (calc II)/Integration by Parts.md +++ b/education/math/MATH1220 (calc II)/Integration by Parts.md @@ -1,6 +1,11 @@ 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: @@ -32,4 +37,6 @@ $$\int xe^{2x}dx$$ 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 - $$ \int\frac{1}{2}e^{w} $$ \ No newline at end of file + $$ \int\frac{1}{2}e^{w}\frac{1}{2}dw = \frac{1}{4}\int e^w dw= \frac{1}{4}e^{2x} $$ +4. Combining everything together, we get: + $$ \int xe^{2x}dx = (\frac{1}{2}e^{2x})(x)- (\frac{1}{4}e^2x) + C$$ \ No newline at end of file