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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**
+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 $4$