Evaluate the given definite trigonometric integral. A) Integral of d(theta)/2+cos(theta) from 0...


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}


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

Become a Study.com member to unlock this answer!

View this answer

Part a

{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}...

See full answer below.

Learn more about this topic:

Integral Calculus: Definition & Applications
Integral Calculus: Definition & Applications


Chapter 12 / Lesson 2

Understand 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.

Related to this Question

Explore our homework questions and answers library