From 444d3196359be6bb82fdb6a0f5d5ba32fde4118a Mon Sep 17 00:00:00 2001
From: Paweł Redman <pawel.redman@gmail.com>
Date: Sun, 14 Jun 2026 18:36:53 +0200
Subject: Initial commit

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 site/content/integrating_sin2x.typ | 66 ++++++++++++++++++++++++++++++++++++++
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 create mode 100644 site/content/integrating_sin2x.typ

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diff --git a/site/content/integrating_sin2x.typ b/site/content/integrating_sin2x.typ
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+#import "/common.typ": post, link-self
+
+#show: post.with(
+    title: "Integrating sin² x",
+    date: datetime(year: 2018, month: 5, day: 1),
+)
+
+$
+  integral sin^2 x dif x = (x - sin x cos x) / 2 + C
+$
+
+= Using the Pythagorean identity
+
+Integrate by parts using $u = sin x, d u = cos x$ and $v = -cos x, d v = sin x$:
+
+$
+  & integral sin x sin x dif x = -sin x cos x - integral -cos x cos x dif x \
+  & integral sin^2 x dif x = -sin x cos x + integral cos^2 x dif x
+$
+
+Add $integral (sin x)^2 dif x$ to both sides:
+
+$
+  2 integral sin^2 x dif x = -sin x cos x + integral cos^2 x dif x + integral sin^2 x dif x
+$
+
+Apply integral linearity:
+
+$
+  2 integral sin^2 x dif x = -sin x cos x + integral (cos^2 x + sin^2 x) dif x
+$
+
+Apply the Pythagorean identity and divide both sides by 2:
+
+$
+  2 integral sin^2 x dif x &= -sin x cos x + integral 1 dif x \
+  &= -sin x cos x + x + C \
+  integral sin^2 x dif x &= (x - sin x cos x) / 2 + C
+$
+
+= Using the double angle formula
+
+#html.elem("aside", attrs: (class: "warning"))[
+  Careful! This derivation differs from the one in #link-self("integrating_cos2x")[integrating_cos2x] by just a sign.
+]
+Rewrite the integral using the double angle formula for the cosine:
+
+$
+  integral sin^2 x dif x = integral (1 - cos 2 x) / 2 dif x
+$
+
+Substitute $u = 2 x, d u = 2 dif x$:
+
+$
+  integral (1 - cos 2 x) / 2 dif x &= integral (1 - cos u) / 2 1 / 2 dif u \
+  &= 1 / 4 (integral 1 dif u - integral cos u dif u) \
+  &= (u - sin u + C) / 4
+$
+
+Substitute back and simplify:
+
+$
+  (u - sin u + C) / 4 &= (2 x - sin 2 x + C) / 4 \
+  &= (x - (sin 2 x) / 2) / 2 + C \
+  &= (x - sin x cos x) / 2 + C
+$
-- 
cgit