Mathematics

Integrate $$\int x\log 2x \, dx$$


SOLUTION
$$I=\displaystyle\int{x\log{2x}dx}$$
Let $$u=\log{2x}\Rightarrow\,du=\dfrac{1}{2x}\times 2\,dx=\dfrac{dx}{x}$$
$$dv=x\,dx\Rightarrow\,v=\dfrac{{x}^{2}}{2}$$
$$I=\dfrac{{x}^{2}}{2}\log{2x}-\displaystyle\int{\dfrac{{x}^{2}}{2}\dfrac{dx}{x}}$$
$$=\dfrac{{x}^{2}}{2}\log{2x}-\dfrac{1}{2}\displaystyle\int{x\,dx}$$
$$=\dfrac{{x}^{2}}{2}\log{2x}-\dfrac{{x}^{2}}{4}+c$$
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Subjective Medium Published on 17th 09, 2020
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