From 97ae37f617cfb2525d4519639200b8f507f54203 Mon Sep 17 00:00:00 2001 From: arc Date: Fri, 5 Sep 2025 12:54:16 -0600 Subject: [PATCH] vault backup: 2025-09-05 12:54:16 --- .../Integration with Trig Identities.md | 15 +++++++++++++-- 1 file changed, 13 insertions(+), 2 deletions(-) diff --git a/education/math/MATH1220 (calc II)/Integration with Trig Identities.md b/education/math/MATH1220 (calc II)/Integration with Trig Identities.md index c5085c6..000a48b 100644 --- a/education/math/MATH1220 (calc II)/Integration with Trig Identities.md +++ b/education/math/MATH1220 (calc II)/Integration with Trig Identities.md @@ -22,7 +22,7 @@ $$ -\int(1 - 2u^2 + u^4)du $$ 6. Substituting $\cos(x)$ back in for $u$, we get the evaluated (but not entirely simplified) integral: $$-(\cos(x)- \frac{2}{3}\cos^3x + \frac{1}{5}\cos^5x) $$ # Trigonometric Substitutions -Trigonometric substitution is useful for equations containing something along the form of $\sqrt{a^2 + x^2}$ or $a^2 + x^2$. +Trigonometric substitution is useful for equations containing $\sqrt{a^2 + x^2}$ or $a^2 + x^2$, where $a$ is any constant. The general process involves the use of a trig identity and multiplying everything in that identity by a constant. @@ -30,4 +30,15 @@ Consider the identity: $$ 1 + \tan^2(\theta) = \sec^2(\theta)$$ Multiplying both sides of the identity by $a^2$, we get: $$a^2 + a^2\tan^2(\theta) = a^2\sec^2(\theta)$$ -This enables us to make use of **sub** \ No newline at end of file +This enables us to make use of **substitution** to simplify many integrals. +- $x = a\tan \theta$ +- $dx = a \sec^2\theta d\theta$ +- for $-\frac{\pi}{2} < \theta < \frac{\pi}{2}$ + +# Examples +> Evaluate the integral $\int\frac{3}{4+x^2}dx$ +1. Move the constant coefficient out of the integral: +$$ \int \frac{3}{4+x^2}dx = 3\int \frac{1}{4 + x^2}dx$$ +2. Let $x = 2tan\theta$ and $dx = (2sec^2\theta d\theta)$, substitute accordingly +$$ = 3\int\frac{1}{4 + 4\tan^2\theta}(2\sec^2\theta)d\theta$$ +3. Factor $4$ \ No newline at end of file