Mathematics

The value of $$\int \frac{x^3}{x+2} dx$$ equals


ANSWER

$$\frac{x^3}{3} -x^2+4x - 8 log |x +2| +c$$


SOLUTION

$$\int (\frac{x^3}{x+2})dx\\Let\> x+2=t\\then \>dx =dt\\\therefore I=\int (\frac{(t-2)^3}{t})dt\\=\int (\frac{t^3-6t^2+12t-8}{t}) dt \\=\int t^2-6t+12-(\frac{8}{t})dt\\=(\frac{t^3}{3}) -(\frac{6t^2}{2}) +12t-8lot +c\\=(\frac{(x+20)^3}{3})-3(x+2)^2+12(x+2) -8log\left | x+2 \right | +c\\=(\frac{x^3}{3}) -x^2+4x-8 log\left | x+2 \right |+c$$

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Single Correct Medium Published on 17th 09, 2020
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