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# Solve the trigonometric integral $\int\frac{1}{\sin\left(x\right)-1}dx$

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asin
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sinh
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asinh
acosh
atanh
acoth
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##  Final answer to the problem

$\frac{2}{\tan\left(\frac{x}{2}\right)-1}+C_0$
Got another answer? Verify it here!

##  Step-by-step Solution 

How should I solve this problem?

• Choose an option
• 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
Can't find a method? Tell us so we can add it.
1

We can solve the integral $\int\frac{1}{\sin\left(x\right)-1}dx$ 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

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

$\sin x=\frac{2t}{1+t^{2}},\:\cos x=\frac{1-t^{2}}{1+t^{2}},\:\mathrm{and}\:\:dx=\frac{2}{1+t^{2}}dt$
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Substituting in the original integral we get

$\int\frac{1}{\frac{2t}{1+t^{2}}-1}\frac{2}{1+t^{2}}dt$
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Simplifying

$\int\frac{2}{2t-\left(1+t^{2}\right)}dt$
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Rewrite the expression $\frac{2}{2t-\left(1+t^{2}\right)}$ inside the integral in factored form

$\int\frac{2}{-\left(t-1\right)^{2}}dt$
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Simplify the division $2$ by $-1$

$\int\frac{-2}{\left(t-1\right)^{2}}dt$
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We can solve the integral $\int\frac{-2}{\left(t-1\right)^{2}}dt$ by applying integration by substitution method (also called U-Substitution). First, we must identify a section within the integral with a new variable (let's call it $u$), which when substituted makes the integral easier. We see that $t-1$ it's a good candidate for substitution. Let's define a variable $u$ and assign it to the choosen part

$u=t-1$
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Now, in order to rewrite $dt$ in terms of $du$, we need to find the derivative of $u$. We need to calculate $du$, we can do that by deriving the equation above

$du=dt$
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Substituting $u$ and $dt$ in the integral and simplify

$\int\frac{-2}{u^{2}}du$
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Rewrite the exponent using the power rule $\frac{a^m}{a^n}=a^{m-n}$, where in this case $m=0$

$\int-2u^{-2}du$
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The integral of a function times a constant ($-2$) is equal to the constant times the integral of the function

$-2\int u^{-2}du$
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Apply the power rule for integration, $\displaystyle\int x^n dx=\frac{x^{n+1}}{n+1}$, where $n$ represents a number or constant function, such as $-2$

$-2\left(\frac{u^{-1}}{-1}\right)$
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Simplify the fraction $-2\left(\frac{u^{-1}}{-1}\right)$

$2u^{-1}$
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Replace $u$ with the value that we assigned to it in the beginning: $t-1$

$2\left(t-1\right)^{-1}$
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Applying the property of exponents, $\displaystyle a^{-n}=\frac{1}{a^n}$, where $n$ is a number

$\frac{2}{t-1}$
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Replace $t$ with the value that we assigned to it in the beginning: $\tan\left(\frac{x}{2}\right)$

$\frac{2}{\tan\left(\frac{x}{2}\right)-1}$
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As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$

$\frac{2}{\tan\left(\frac{x}{2}\right)-1}+C_0$

##  Final answer to the problem

$\frac{2}{\tan\left(\frac{x}{2}\right)-1}+C_0$

##  Explore different ways to solve this problem

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

SnapXam A2

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0
a
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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

###  Main Topic: Trigonometric Integrals

Integrals that contain trigonometric functions and their powers.