Mathematics

Evaluate $$\displaystyle \int { \cfrac { { x }^{ 3 }+1 }{ { x }^{ 2 }+1 }  } dx=$$


SOLUTION
$$\displaystyle\int \dfrac{x^{3}+1}{x^{2}+1}dx=\int \dfrac{(x+1)(x^{2}+1-x)}{(x^{2}+1)}$$
$$=\displaystyle\int (1-\dfrac{x}{x^{2}+1})(x+1)$$
$$=\displaystyle\int (x+1)dx-\int \dfrac{x^{2}+x}{x^{2}+1}dx$$
$$=\displaystyle\int (x+1)dx-\int dx-\int \dfrac{x-1}{x^{2}+1}dx$$
$$=\displaystyle\int x.dx-\frac{1}{2}\int \dfrac{2x}{x^{2}+1}dx+\int \dfrac{1}{1+x^{2}}dx$$
$$=\dfrac{x^{2}}{2}-\dfrac{1}{2}dx(1+x^{2})+\tan^{-1}x+c$$
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Subjective Medium Published on 17th 09, 2020
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