Mathematics

Find $$\displaystyle \int \dfrac{x^4+1}{x(x^2+1)^2}$$


SOLUTION
Given $$\int{\dfrac{x^4+1}{x(x^2+1)^2}dx}$$
$$=\int{\left(\dfrac{1}{x}-\dfrac{2x}{(x^2+1)^2}\right)dx}$$
$$=\int{\dfrac{1}{x}dx}-2\int{\dfrac{x}{(x^2+1)^2}dx}$$
$$\int{\dfrac{1}{x}dx}=ln(x)$$
$$\int{\dfrac{x}{(x^2+1)^2dx}}$$
Substituting $$u=x^2+1$$    $$dx=\dfrac{1}{2x}du$$
$$=\dfrac{1}{2}\int{\dfrac{1}{u^2}du}$$
$$=-\dfrac{1}{2u}$$
$$=-\dfrac{1}{2(x^2+1)}$$
Putting in solved integrals
$$=\int{\dfrac{1}{x}dx}-2\int{\dfrac{x}{(x^2+1)^2}dx}$$
$$=ln(x)+\dfrac{1}{x^2+1}+C$$
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Subjective Medium Published on 17th 09, 2020
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