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Chapter 7: Problem 83

Refer to the summary box (Partial Fraction Decompositions) and evaluate the following integrals. $$\int \frac{2}{x\left(x^{2}+1\right)^{2}} d x$$

Short Answer

Expert verified
$$\int \frac{2}{x\left(x^{2}+1\right)^{2}} d x$$ The integral of the given function is: $$\int \frac{2}{x(x^2+1)^2} dx = \frac{1}{2}\ln{\frac{x^2+1}{x}} - \frac{1}{2}\arctan{x} - \frac{1}{2}\frac{x}{x^2+1} + C$$

Step by step solution

01

Decompose the integrand into partial fractions

We are given the integral: $$\int \frac{2}{x\left(x^{2}+1\right)^{2}} d x$$ First, decompose the given integrand into partial fractions. We can write it as: $$\frac{2}{x(x^2+1)^2} = \frac{A}{x} + \frac{Bx+C}{x^2+1} + \frac{Dx+E}{(x^2+1)^2}$$ Next, clear the denominators by multiplying both sides by \(x(x^2 + 1)^2\). This gives us: $$2 = A(x^2+1)^2 + (Bx+C)x(x^2+1) + (Dx+E)x$$
02

Equate coefficients and solve the system of equations

Now, we equate the coefficients for the polynomial on both sides of the equation. This gives the following system of equations: $$ \begin{cases} A + B = 0 \\ 2A + C = 0 \\ 2B + D = 0 \\ 2A + E = 2 \end{cases} $$ Solving this system, we get \(A=-B, C=2A, D=-2B, E=2A\). By solving for \(A\) and \(B\), we find their values to be \(A=-\frac{1}{2}\) and \(B=\frac{1}{2}\). Then we can find \(C=-1, D=1, E=-1\). Now we can express the integrand in terms of its partial fractions: $$\frac{2}{x(x^2+1)^2} = -\frac{1}{2x} + \frac{x-1}{x^2+1} + \frac{x-1}{(x^2+1)^2}$$
03

Integrate each partial fraction separately

We now have the integral as the sum of three separate integrals: $$\int \frac{2}{x(x^2+1)^2} dx = -\frac{1}{2}\int\frac{1}{x} dx + \int\frac{x-1}{x^2+1} dx + \int\frac{x-1}{(x^2+1)^2}dx$$ Evaluate these integrals: 1. \(-\frac{1}{2}\int\frac{1}{x} dx = -\frac{1}{2}\ln|x| + C_1\) 2. \(\int\frac{x-1}{x^2+1} dx = \frac{1}{2}\ln|x^2+1| - \arctan{x} + C_2\) 3. \(\int\frac{x-1}{(x^2+1)^2}dx = \frac{1}{2}\int\frac{1}{1+x^2} dx - \int\frac{1}{(x^2+1)^2}dx = \frac{1}{2}\arctan{x} - \frac{1}{2}(x(x^2 + 1)^{-1}) + C_3\)
04

Combine the results and write the final answer

Sum up the results of these integrals: $$\int \frac{2}{x(x^2+1)^2} dx = -\frac{1}{2}\ln|x| + \frac{1}{2}\ln|x^2+1| - \arctan{x} + \frac{1}{2}\arctan{x} - \frac{1}{2}\frac{x}{x^2+1} + C$$ Simplify it: $$\int \frac{2}{x(x^2+1)^2} dx = \frac{1}{2}\ln{\frac{x^2+1}{x}} - \frac{1}{2}\arctan{x} - \frac{1}{2}\frac{x}{x^2+1} + C$$ This is the final result.

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