From 3529515066c6077b40dabe7d5db8809335e93869 Mon Sep 17 00:00:00 2001 From: arc Date: Fri, 5 Sep 2025 12:49:15 -0600 Subject: [PATCH] vault backup: 2025-09-05 12:49:15 --- .../Integration with Trig Identities.md | 12 +++++++++++- 1 file changed, 11 insertions(+), 1 deletion(-) 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 9831767..c5085c6 100644 --- a/education/math/MATH1220 (calc II)/Integration with Trig Identities.md +++ b/education/math/MATH1220 (calc II)/Integration with Trig Identities.md @@ -20,4 +20,14 @@ $$ -\int(1 - 2u^2 + u^4)du $$ 5. Take advantage of the distributive property of integrals: $$ - (u - \frac{2}{3}u^3 + \frac{1}{5}u^5) + C $$ 6. Substituting $\cos(x)$ back in for $u$, we get the evaluated (but not entirely simplified) integral: - $$-(\cos(x)- \frac{2}{3}\cos^3x + \frac{1}{5}\cos^5x) $$ \ No newline at end of file + $$-(\cos(x)- \frac{2}{3}\cos^3x + \frac{1}{5}\cos^5x) $$ +# Trigonometric Substitutions +Trigonometric substitution is useful for equations containing something along the form of $\sqrt{a^2 + x^2}$ or $a^2 + x^2$. + +The general process involves the use of a trig identity and multiplying everything in that identity by a constant. + +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