Mathematics

Evaluate: $$\displaystyle \int{\dfrac{{x}^{3}}{x+2}dx}$$


SOLUTION
$$ \int  \frac{x^{3}dx}{x+2} = \int \frac{(x^{3}+8-8)dx}{x+2}$$
$$ =\int  \frac{(x^{3}+8)dx}{(x+2)} - \int \frac{8dx}{x+2}$$
$$ =\int \frac{(x+2)(x^{2}-2x+4)dx}{(x+2)} -8 \int \frac{dx}{x+2}$$
$$ = \int (x^{2}-2x+4)dx - 8 \int \frac{dx}{x+2}$$
$$ = \frac{x^{3}}{3}-\frac{2x^{2}}{2}+4x-8log |x+2|+c$$
$$ = \frac{x^{3}}{3} -x^{2}+4x-8 log |x+2| +c$$
Where c is the constant of integration
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Subjective Medium Published on 17th 09, 2020
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