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

Step-by-step Solution

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

Specify the solving method

1

We can solve the integral $\int\frac{x}{x^2-1}dx$ 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 $x^2-1$ it's a good candidate for substitution. Let's define a variable $u$ and assign it to the choosen part

$u=x^2-1$

Differentiate both sides of the equation $u=x^2-1$

$du=\frac{d}{dx}\left(x^2-1\right)$

Find the derivative

$\frac{d}{dx}\left(x^2-1\right)$

The derivative of a sum of two or more functions is the sum of the derivatives of each function

$\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(-1\right)$

The derivative of the constant function ($-1$) is equal to zero

$\frac{d}{dx}\left(x^2\right)$

The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$

$2x$
2

Now, in order to rewrite $dx$ 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

$du=2xdx$
3

Isolate $dx$ in the previous equation

$\frac{du}{2x}=dx$

Simplify the fraction $\frac{\frac{x}{u}}{2x}$ by $x$

$\int\frac{1}{2u}du$

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

$\frac{1}{2}\int\frac{1}{u}du$
4

Substituting $u$ and $dx$ in the integral and simplify

$\frac{1}{2}\int\frac{1}{u}du$
5

The integral of the inverse of the lineal function is given by the following formula, $\displaystyle\int\frac{1}{x}dx=\ln(x)$

$\frac{1}{2}\ln\left(u\right)$

$\frac{1}{2}\ln\left(x^2-1\right)$
6

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

$\frac{1}{2}\ln\left(x^2-1\right)$
7

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

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

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

 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

Main Topic: Integrals of Rational Functions

Integrals of rational functions of the form R(x) = P(x)/Q(x).

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