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education/math/MATH1220 (calc II)/Integration by Parts.md

@@ -23,4 +23,13 @@ $$\int xe^{2x}dx$$
 	- $dv = e^{2x}dx$
 	- $v = \frac{1}{2}e^{2x}$ - The antiderivative of $dv$.
 2. Looking back at the integration by parts formula, we know that:
-	$$ \int udv = uv - \int v du $$
+	$$ \int udv = uv - \int v du $$
+	$$ \int xe^{2x}dx = (\frac{1}{2}e^{2x})(x)-\int (\frac{1}{2}e^{2x}) (1dx) $$
+3. The remaining integral can be solved with $u$ substitution, but we've already defined $u$, so we use $w$ as a replacement.
+	- $w = 2x$
+	- $dw = 2dx$
+	- $\frac{1}{2}dw=dx$ 
+		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
+		$$ \int\frac{1}{2}e^{w} $$