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
$4x=A(x^{2}+1)+(Bx+C)(x+1).$

Step 2:
If we let  $x=-1,$  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,  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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 (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle C=-A=2.}
Finally, if we let  $x=1,$  we get  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 4=2A+2B+2C.}
Plugging in  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle A=-2}   and  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle C=2,}   we get  $B=2.$
So, in summation, we have
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 {-2}{x+1}}+{\frac {2x+2}{x^{2}+1}}.}

Step 3:
Now, we have
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\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  $u$-substitution.
For the first integral, we substitute  $u=x+1.$
For the second integral, the substitution is  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle t=x^{2}+1.}
Then, we integrate to get
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\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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