Step-by-step Solution

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

Go!
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Final Answer

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

Problem to solve:

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

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 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

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

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

Multiply $\frac{1}{2}$ times $1$

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

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

Final Answer

$\frac{1}{2}\ln\left(x^2-1\right)+C_0$
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5
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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

Tips on how to improve your answer:

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

Related Formulas:

1. See formulas

Time to solve it:

~ 0.03 s