Mathematics

Solve $$ \int { (\log_ e x)^2} dx $$ o


ANSWER

$$x[(\log x)^2 - 2(\log x)+2] + c$$


SOLUTION
$$\displaystyle I=\int (\log x)^{2}dx$$
Let $$u=(\log (x))^{2}$$         $$du=\dfrac{210gx}{x}dx$$
$$dv=dx$$             $$v=x$$
Now $$\displaystyle \int \log x dx = x \log |x|-x+c$$
$$\therefore \displaystyle \int (\log x dx)^{2}dx=\int udv$$
$$\displaystyle =uv-\int udv$$
$$=\displaystyle x(\log |x|)^{2}-\int 2 \log x dx$$
$$=x(\log |x|)^{2}-2(x \log |x|-x)+c$$
$$=x(\log |x|)^{2}-2x \log |x|+2x+c$$
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Single Correct Medium Published on 17th 09, 2020
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