Integrate the function $\left(35+18\cos\left(x\right)\right)^{-1}$ from 0 to $1$

Step-by-step Solution

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Final answer to the problem

$\frac{2\sqrt{\frac{53}{17}}\arctan\left(\frac{\sqrt{17}\tan\left(\frac{1}{2}\right)}{\sqrt{53}}\right)}{53}- \frac{2\sqrt{\frac{53}{17}}\arctan\left(\frac{\sqrt{17}\tan\left(\frac{0}{2}\right)}{\sqrt{53}}\right)}{53}$
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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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Applying the property of exponents, $\displaystyle a^{-n}=\frac{1}{a^n}$, where $n$ is a number

Learn how to solve integrals of rational functions of sine and cosine problems step by step online.

$\int_{0}^{1}\frac{1}{35+18\cos\left(x\right)}dx$

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Learn how to solve integrals of rational functions of sine and cosine problems step by step online. Integrate the function (35+18cos(x))^(-1) from 0 to 1. Applying the property of exponents, \displaystyle a^{-n}=\frac{1}{a^n}, where n is a number. We can solve the integral \int\frac{1}{35+18\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.

Final answer to the problem

$\frac{2\sqrt{\frac{53}{17}}\arctan\left(\frac{\sqrt{17}\tan\left(\frac{1}{2}\right)}{\sqrt{53}}\right)}{53}- \frac{2\sqrt{\frac{53}{17}}\arctan\left(\frac{\sqrt{17}\tan\left(\frac{0}{2}\right)}{\sqrt{53}}\right)}{53}$

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Function Plot

Plotting: $\left(35+18\cos\left(x\right)\right)^{-1}$

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0
a
b
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f
g
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: Integrals of Rational Functions of Sine and Cosine

The integrals of rational functions of sine and cosine are integrals where the integrand is a rational function composed of sines and cosines, usually in the denominator, which is not easily reducible. However, these integrals are easy to solve by applying the recommended substitution: $z=\tan\left(\frac{x}{2}\right)$

Used Formulas

See formulas (3)

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