diff options
Diffstat (limited to 'site/content/integrating_cos2x.typ')
| -rw-r--r-- | site/content/integrating_cos2x.typ | 67 |
1 files changed, 0 insertions, 67 deletions
diff --git a/site/content/integrating_cos2x.typ b/site/content/integrating_cos2x.typ deleted file mode 100644 index a7cfe4b..0000000 --- a/site/content/integrating_cos2x.typ +++ /dev/null @@ -1,67 +0,0 @@ -#import "/common.typ": post, link-self - -#show: post.with( - title: "Integrating cos² x", - date: datetime(year: 2018, month: 5, day: 1), -) - -$ - integral cos^2 x dif x = (x + sin x cos x) / 2 + C -$ - -= Using the Pythagorean identity - -Integrate by parts using $u = cos x, d u = -sin x$ and $v = sin x, d v = cos x$: - -$ - & integral cos x cos x dif x = sin x cos x - integral -sin x sin x dif x \ - & integral cos^2 x dif x = sin x cos x + integral sin^2 x dif x -$ - -Add $integral (cos x)^2 dif x$ to both sides: - -$ - 2 integral cos^2 x dif x = sin x cos x + integral sin^2 x dif x + integral cos^2 x dif x -$ - -Apply integral linearity: - -$ - 2 integral cos^2 x dif x = sin x cos x + integral (sin^2 x + cos^2 x) dif x -$ - -Apply the Pythagorean identity and divide both sides by 2: - -$ - 2 integral cos^2 x dif x &= sin x cos x + integral 1 dif x \ - & = sin x cos x + x + C \ - integral cos^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_sin2x")[integrating_sin2x] by just a sign. -] - -Rewrite the integral using the double angle formula for the cosine: - -$ - integral cos^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 -$ |