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

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Solution

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Final Answer

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

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

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

Learn how to solve integrals of rational functions problems step by step online.

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

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

Final Answer

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

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

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Function Plot

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

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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
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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Main Topic: Integrals of Rational Functions

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

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