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# Find the integral $\int\frac{x}{x^2-1}dx$

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##  Final answer to the problem

$\frac{1}{2}\ln\left|x^2-1\right|+C_0$
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##  Step-by-step Solution 

How should I solve this problem?

• Integrate by trigonometric substitution
• Integrate by partial fractions
• Integrate by substitution
• Integrate by parts
• Integrate using tabular integration
• Weierstrass Substitution
• Integrate using trigonometric identities
• Integrate using basic integrals
• Product of Binomials with Common Term
• FOIL Method
• Load more...
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1

We can solve the integral $\int\frac{x}{x^2-1}dx$ by applying integration method of trigonometric substitution using the substitution

$x=\sec\left(\theta \right)$
2

Now, in order to rewrite $d\theta$ in terms of $dx$, we need to find the derivative of $x$. We need to calculate $dx$, we can do that by deriving the equation above

$dx=\sec\left(\theta \right)\tan\left(\theta \right)d\theta$
3

Substituting in the original integral, we get

$\int\frac{\sec\left(\theta \right)^2\tan\left(\theta \right)}{\sec\left(\theta \right)^2-1}d\theta$
4

Apply the trigonometric identity: $\sec\left(\theta \right)^2-1$$=\tan\left(\theta \right)^2$, where $x=\theta$

$\int\frac{\sec\left(\theta \right)^2\tan\left(\theta \right)}{\tan\left(\theta \right)^2}d\theta$
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5

Simplify the fraction by $\tan\left(\theta \right)$

$\int\frac{\sec\left(\theta \right)^2}{\tan\left(\theta \right)}d\theta$
6

Rewrite the trigonometric expression $\frac{\sec\left(\theta \right)^2}{\tan\left(\theta \right)}$ inside the integral

$\int\sec\left(\theta \right)\csc\left(\theta \right)d\theta$
7

Reduce $\sec\left(\theta \right)\csc\left(\theta \right)$ by applying trigonometric identities

$\int\frac{\csc\left(\theta \right)}{\cos\left(\theta \right)}d\theta$
8

Rewrite the trigonometric expression $\frac{\csc\left(\theta \right)}{\cos\left(\theta \right)}$ inside the integral

$\int2\csc\left(2\theta \right)d\theta$
9

The integral of a function times a constant ($2$) is equal to the constant times the integral of the function

$2\int\csc\left(2\theta \right)d\theta$
10

We can solve the integral $\int\csc\left(2\theta \right)d\theta$ 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 $2\theta$ it's a good candidate for substitution. Let's define a variable $u$ and assign it to the choosen part

$u=2\theta$
11

Now, in order to rewrite $d\theta$ 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

$du=2d\theta$
12

Isolate $d\theta$ in the previous equation

$d\theta=\frac{du}{2}$
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Substituting $u$ and $d\theta$ in the integral and simplify

$2\int\frac{\csc\left(u\right)}{2}du$
14

Take the constant $\frac{1}{2}$ out of the integral

$2\cdot \left(\frac{1}{2}\right)\int\csc\left(u\right)du$
15

Multiply the fraction and term in $2\cdot \left(\frac{1}{2}\right)\int\csc\left(u\right)du$

$\int\csc\left(u\right)du$
16

The integral of $\csc(x)$ is $-\ln(\csc(x)+\cot(x))$

$-\ln\left|\csc\left(u\right)+\cot\left(u\right)\right|$
17

Replace $u$ with the value that we assigned to it in the beginning: $2\theta$

$-\ln\left|\csc\left(2\theta \right)+\cot\left(2\theta \right)\right|$
18

Simplify $\csc\left(2\theta \right)+\cot\left(2\theta \right)$ using trigonometric identities

$-\ln\left|\cot\left(\theta \right)\right|$
19

Express the variable $\theta$ in terms of the original variable $x$

$\frac{1}{2}\ln\left|x^2-1\right|$
20

As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$

$\frac{1}{2}\ln\left|x^2-1\right|+C_0$

##  Final answer to the problem

$\frac{1}{2}\ln\left|x^2-1\right|+C_0$

##  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

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1
2
3
4
5
6
7
8
9
0
a
b
c
d
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

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