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

Step-by-step Solution

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Solution

Final Answer

$x+\frac{3}{2}\ln\left(x-1\right)-\frac{3}{2}\ln\left(x+1\right)+\frac{1}{2}\ln\left(x^2-1\right)+C_0$
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Step-by-step Solution

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Divide $x^2+x+2$ by $x^2-1$

$\begin{array}{l}\phantom{\phantom{;}x^{2}-1;}{\phantom{;}1\phantom{;}\phantom{;}}\\\phantom{;}x^{2}-1\overline{\smash{)}\phantom{;}x^{2}+x\phantom{;}+2\phantom{;}\phantom{;}}\\\phantom{\phantom{;}x^{2}-1;}\underline{-x^{2}\phantom{-;x^n}+1\phantom{;}\phantom{;}}\\\phantom{-x^{2}+1\phantom{;}\phantom{;};}\phantom{;}x\phantom{;}+3\phantom{;}\phantom{;}\\\end{array}$

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$\begin{array}{l}\phantom{\phantom{;}x^{2}-1;}{\phantom{;}1\phantom{;}\phantom{;}}\\\phantom{;}x^{2}-1\overline{\smash{)}\phantom{;}x^{2}+x\phantom{;}+2\phantom{;}\phantom{;}}\\\phantom{\phantom{;}x^{2}-1;}\underline{-x^{2}\phantom{-;x^n}+1\phantom{;}\phantom{;}}\\\phantom{-x^{2}+1\phantom{;}\phantom{;};}\phantom{;}x\phantom{;}+3\phantom{;}\phantom{;}\\\end{array}$

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Learn how to solve problems step by step online. Find the integral int((x^2+x+2)/(x^2-1))dx. Divide x^2+x+2 by x^2-1. Resulting polynomial. Expand the integral \int\left(1+\frac{x+3}{x^2-1}\right)dx into 2 integrals using the sum rule for integrals, to then solve each integral separately. The integral \int1dx results in: x.

Final Answer

$x+\frac{3}{2}\ln\left(x-1\right)-\frac{3}{2}\ln\left(x+1\right)+\frac{1}{2}\ln\left(x^2-1\right)+C_0$

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

Solve integral of ((x^2+x)/(x^2-1))dx using partial fractionsSolve integral of ((x^2+x)/(x^2-1))dx using basic integralsSolve integral of ((x^2+x)/(x^2-1))dx using u-substitutionSolve integral of ((x^2+x)/(x^2-1))dx using integration by partsSolve integral of ((x^2+x)/(x^2-1))dx using trigonometric substitution

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Function Plot

Plotting: $x+\frac{3}{2}\ln\left(x-1\right)-\frac{3}{2}\ln\left(x+1\right)+\frac{1}{2}\ln\left(x^2-1\right)+C_0$

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1
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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
D□x
∫
∫◻
|◻|
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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