Evaluate the given definite trigonometric integral. A) Integral of d(theta)/2+cos(theta) from 0...
Question:
Evaluate the given definite trigonometric integral.
A) {eq}\int_{0}^{2\pi}\frac{d\theta}{2+cos\theta } {/eq}
B) {eq}\int_{0}^{2\pi}\frac{d\theta}{3+sin\theta +cos\theta } {/eq}
Integral
To solve these integrals apply the rational substitution
{eq}\tan(\theta/2) = x,\;\;\; \cos(\theta) =\frac{1-x^2}{1+x^2},\;\;\; \sin(\theta) =\frac{2x}{1+x^2},\;\;\; d\theta =\frac{2\, dx}{1+x^2} {/eq}
and convert the trigonometric integral to a partial fraction problem.
Answer and Explanation: 1
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{eq}\begin{align*} \int_{0}^{2\pi}\frac{d\theta}{2+cos\theta } &=2\int_{0}^{\pi}\frac{d\theta}{2+cos\theta } \\ &= 2\int_{\infty}^{0}...
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Chapter 12 / Lesson 2Understand that an integral measures the area under a curve, and learn how to evaluate linear and polynomial integrals. Explore different applications of integrals with examples.