Mathematics

$$ \int x.log(x+1)dx$$


SOLUTION
Consider the given integral.
$$I=\displaystyle \int x.log(x+1)dx$$
$$I=log(x+1) .\dfrac{x^2}{2}-\int \dfrac{1}{(x+1)}\dfrac{x^2}{2}dx$$
$$I=log(x+1) .\dfrac{x^2}{2}-\int \dfrac{1}{(x+1)}\dfrac{(x^2-1+1)}{2}dx$$
$$I=log(x+1) .\dfrac{x^2}{2}-\dfrac{1}{2} \displaystyle \int \left[ (x-1)+\dfrac{1}{x+1}\right] dx$$
$$I=log(x+1) .\dfrac{x^2}{2}-\dfrac{1}{2} \left[ \dfrac{x^2}{2}-x+log(x+1)\right]+C$$
where $$c$$ is the constant of integration.
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Subjective Medium Published on 17th 09, 2020
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