Mathematics

$$I=\displaystyle \int \dfrac{(x+a)^3}{x^3}dx $$ is equal to:


ANSWER

$$x+ 3a \log x -\dfrac{3a^2}{x} - \dfrac{a^3}{2x^2}+c$$


SOLUTION
$$I=\displaystyle \int \dfrac{(x+a)^3}{x^3}dx $$

$$=\displaystyle \int \dfrac {x^{3}+a^{3}+3ax^{2}+3a^{2}x}{x^{3}}dx$$

$$=\displaystyle \int \left (1+\dfrac {a^{3}}{x^{3}}+\dfrac {3a}{x}+\dfrac {3a^{2}}{x^{2}}\right) dx$$

$$=x-\dfrac {a^{3}}{2x^{2}}+3a\log x - \dfrac {3a^{2}}{x}+c$$

$$=x+3a\log x - \dfrac {3a^{2}}{x}-\dfrac {a^{3}}{2x^{2}}+c$$
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Single Correct Medium Published on 17th 09, 2020
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