em dash; sentence case

quarto/integrals/integration_by_parts.qmd

@@ -1,4 +1,4 @@
-# Integration By Parts
+# Integration by parts
 
 
 {{< include ../_common_code.qmd >}}
@@ -116,7 +116,7 @@ $$
 
 $B$ is similar with the roles of $u$ and $v$ reversed.
 
-----
+---
 
 Informally, the integration by parts formula is sometimes seen as $\int udv = uv - \int v du$, as well can be somewhat confusingly written as:
 
@@ -382,7 +382,7 @@ Recall, just using *either* $x_i$ or $x_{i-1}$ for $c_i$ gives an error that is
 This [proof](http://www.math.ucsd.edu/~ebender/20B/77_Trap.pdf) for the error estimate is involved, but is reproduced here, as it nicely integrates many of the theoretical concepts of integration discussed so far.
 
 
-First, for convenience, we consider the interval $x_i$ to $x_i+h$. The actual answer over this is just $\int_{x_i}^{x_i+h}f(x) dx$. By a $u$-substitution with $u=x-x_i$ this becomes $\int_0^h f(t + x_i) dt$. For analyzing this we integrate once by parts using $u=f(t+x_i)$ and $dv=dt$. But instead of letting $v=t$, we choose to add--as is our prerogative--a constant of integration $A$, so $v=t+A$:
+First, for convenience, we consider the interval $x_i$ to $x_i+h$. The actual answer over this is just $\int_{x_i}^{x_i+h}f(x) dx$. By a $u$-substitution with $u=x-x_i$ this becomes $\int_0^h f(t + x_i) dt$. For analyzing this we integrate once by parts using $u=f(t+x_i)$ and $dv=dt$. But instead of letting $v=t$, we choose to add---as is our prerogative---a constant of integration $A$, so $v=t+A$:
 
 
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