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

Step-by-step Solution

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Solution

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 $\int\frac{1}{2+\cos\left(a\right)}da$ 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

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

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Learn how to solve problems step by step online. Integrate the function 1/(2+cos(a)) from 0 to 2*pi. We can solve the integral \int\frac{1}{2+\cos\left(a\right)}da 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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Solving a math problem using different methods is important because it enhances understanding, encourages critical thinking, allows for multiple solutions, and develops problem-solving strategies. Read more

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Function Plot

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

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0
a
b
c
d
f
g
m
n
u
v
w
x
y
z
.
(◻)
+
-
×
◻/◻
/
÷
◻2
◻◻
√◻
√

◻√◻
◻√
∞
e
π
ln
log
log◻
lim
d/dx
D□x
∫
∫◻
|◻|
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

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