Integrate the function $\frac{1}{2+\cos\left(x\right)}$ from 0 to $2\pi $

Step-by-step Solution

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Final answer to the problem

$2\cdot \left(\frac{1}{\sqrt{3}}\right)\arctan\left(\frac{\tan\left(\frac{2\pi }{2}\right)}{\sqrt{3}}\right)- 2\cdot \left(\frac{1}{\sqrt{3}}\right)\arctan\left(\frac{\tan\left(\frac{0}{2}\right)}{\sqrt{3}}\right)$
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Step-by-step Solution

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  • Integrate by partial fractions
  • Integrate by substitution
  • Integrate by parts
  • Integrate using tabular integration
  • Integrate by trigonometric substitution
  • Weierstrass Substitution
  • Integrate using trigonometric identities
  • Integrate using basic integrals
  • Product of Binomials with Common Term
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We can solve the integral $\int\frac{1}{2+\cos\left(x\right)}dx$ by applying the method Weierstrass substitution (also known as tangent half-angle substitution) which converts an integral of trigonometric functions into a rational function of $t$ by setting the substitution

$t=\tan\left(\frac{x}{2}\right)$

Learn how to solve definite integrals problems step by step online.

$t=\tan\left(\frac{x}{2}\right)$

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Learn how to solve definite integrals problems step by step online. Integrate the function 1/(2+cos(x)) from 0 to 2*pi. We can solve the integral \int\frac{1}{2+\cos\left(x\right)}dx by applying the method Weierstrass substitution (also known as tangent half-angle substitution) which converts an integral of trigonometric functions into a rational function of t by setting the substitution. Hence. Substituting in the original integral we get. Simplifying.

Final answer to the problem

$2\cdot \left(\frac{1}{\sqrt{3}}\right)\arctan\left(\frac{\tan\left(\frac{2\pi }{2}\right)}{\sqrt{3}}\right)- 2\cdot \left(\frac{1}{\sqrt{3}}\right)\arctan\left(\frac{\tan\left(\frac{0}{2}\right)}{\sqrt{3}}\right)$

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Function Plot

Plotting: $\frac{1}{2+\cos\left(x\right)}$

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0
a
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m
n
u
v
w
x
y
z
.
(◻)
+
-
×
◻/◻
/
÷
2

e
π
ln
log
log
lim
d/dx
Dx
|◻|
θ
=
>
<
>=
<=
sin
cos
tan
cot
sec
csc

asin
acos
atan
acot
asec
acsc

sinh
cosh
tanh
coth
sech
csch

asinh
acosh
atanh
acoth
asech
acsch

How to improve your answer:

Main Topic: Definite Integrals

Given a function f(x) and the interval [a,b], the definite integral is equal to the area that is bounded by the graph of f(x), the x-axis and the vertical lines x=a and x=b

Used Formulas

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