007B Sample Midterm 2, Problem 5 Detailed Solution

Evaluate the integral:

$\int {\frac {4x}{(x+1)(x^{2}+1)}}~dx$

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Background Information:
Through partial fraction decomposition, we can write
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\frac {1}{(x+1)(x^{2}+1)}}={\frac {A}{x+1}}+{\frac {Bx+C}{x^{2}+1}}}
for some constants $A,B.$

Solution:

Step 1:
We need to use partial fraction decomposition for this integral.
To start, we let
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\frac {4x}{(x+1)(x^{2}+1)}}={\frac {A}{x+1}}+{\frac {Bx+C}{x^{2}+1}}.}
Multiplying both sides of the last equation by  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (x+1)(x^{2}+1),}
we get
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 4x=A(x^{2}+1)+(Bx+C)(x+1).}

Step 2:
If we let  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle x=-1,}   the last equation becomes  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle -4=2A.}   So,  $A=-2.$
If we let  $x=0,$  then we get  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 0=A+C.}   Thus,  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C=-A=2.}
Finally, if we let  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=1,}   we get  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 4=2A+2B+2C.}
Plugging in  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A=-2}   and  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C=2,}   we get  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B=2.}
So, in summation, we have
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{4x}{(x+1)(x^2+1)}=\frac{-2}{x+1}+\frac{2x+2}{x^2+1}.}

Step 3:
Now, we have
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rcl} \displaystyle{\int \frac{4x}{(x+1)(x^2+1)}~dx} & = & \displaystyle{\int \frac{-2}{x+1}~dx +\int \frac{2x+2}{x^2+1}~dx}\\ &&\\ & = & \displaystyle{\int \frac{-2}{x+1}~dx +\int \frac{2x}{x^2+1}~dx+\int \frac{2}{x^2+1}~dx}\\ &&\\ & = & \displaystyle{\int \frac{-2}{x+1}~dx +\int \frac{2x}{x^2+1}~dx+2\arctan(x).} \end{array}}
For the remaining integrals, we use  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u} -substitution.
For the first integral, we substitute  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u=x+1.}
For the second integral, the substitution is  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t=x^2+1.}
Then, we integrate to get
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rcl} \displaystyle{\int \frac{4x}{(x+1)(x^2+1)}~dx} & = & \displaystyle{\int \frac{-2}{u}~du +\int \frac{1}{t}~dt+2\arctan(x)}\\ &&\\ & = & \displaystyle{-2 \ln |u| +\ln |t|+2\arctan(x)+C}\\ &&\\ & = & \displaystyle{-2 \ln |x+1|+\ln |x^2+1|+2\arctan(x)+C.} \end{array}}

Final Answer:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -2 \ln |x+1|+\ln |x^2+1|+2\arctan(x)+C}

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