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education/math/MATH1220 (calc II)/Integration with Trig Identities.md
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@ -20,4 +20,14 @@ $$ -\int(1 - 2u^2 + u^4)du $$
5. Take advantage of the distributive property of integrals: 5. Take advantage of the distributive property of integrals:
$$ - (u - \frac{2}{3}u^3 + \frac{1}{5}u^5) + C $$ $$ - (u - \frac{2}{3}u^3 + \frac{1}{5}u^5) + C $$
6. Substituting $\cos(x)$ back in for $u$, we get the evaluated (but not entirely simplified) integral: 6. Substituting $\cos(x)$ back in for $u$, we get the evaluated (but not entirely simplified) integral:
$$-(\cos(x)- \frac{2}{3}\cos^3x + \frac{1}{5}\cos^5x) $$ $$-(\cos(x)- \frac{2}{3}\cos^3x + \frac{1}{5}\cos^5x) $$
# 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**