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Integrate the function $\frac{x^2}{\left(2+x^2\right)^2}$ from $-1$ to $\infty $

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Solution

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Final Answer

$\frac{\sqrt{2}}{4}\arctan\left(0.707106x\right)+\frac{-x}{2\left(2+x^2\right)}$
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Step-by-step Solution

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Rewrite the fraction $\frac{x^2}{\left(2+x^2\right)^2}$ in $2$ simpler fractions using partial fraction decomposition

$\frac{x^2}{\left(2+x^2\right)^2}=\frac{Ax+B}{2+x^2}+\frac{Cx+D}{\left(2+x^2\right)^{2}}$

Learn how to solve polynomial factorization problems step by step online.

$\frac{x^2}{\left(2+x^2\right)^2}=\frac{Ax+B}{2+x^2}+\frac{Cx+D}{\left(2+x^2\right)^{2}}$

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Learn how to solve polynomial factorization problems step by step online. Integrate the function (x^2)/((2+x^2)^2) from -1 to infinity. Rewrite the fraction \frac{x^2}{\left(2+x^2\right)^2} in 2 simpler fractions using partial fraction decomposition. Find the values for the unknown coefficients: A, B, C, D. The first step is to multiply both sides of the equation from the previous step by \left(2+x^2\right)^2. Multiplying polynomials. Simplifying.

Final Answer

$\frac{\sqrt{2}}{4}\arctan\left(0.707106x\right)+\frac{-x}{2\left(2+x^2\right)}$

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

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Function Plot

Plotting: $\frac{x^2}{\left(2+x^2\right)^2}$

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◻√◻
◻√
∞
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: Polynomial Factorization

They are a group of techniques that help us rewrite polynomial expressions as a product of factors.

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