Mathematics

Evaluate the given integral.
$$\int { { x }^{ 3 }\log { x }  } dx$$


SOLUTION
$$I=\displaystyle\int{{x}^{3}\log{x}dx}$$

Integrating by parts, we get

Let $$dv={x}^{3}dx\Rightarrow\,v=\dfrac{{x}^{4}}{4}$$

$$u=\log{x}\Rightarrow\,du=\dfrac{1}{x}dx$$

$$\int u.v dx=u \int vdx-\int \left [\int vdx. \dfrac{du}{dx}.dx \right ] $$......by parts formula.

$$\Rightarrow\,I=\dfrac{{x}^{4}\log{x}}{4}-\displaystyle\int{\dfrac{{x}^{4}}{4}\times\dfrac{1}{x}dx}$$

$$\Rightarrow\,I=\dfrac{{x}^{4}\log{x}}{4}-\dfrac{1}{4}\displaystyle\int{{x}^{3}dx}$$

$$\Rightarrow\,I=\dfrac{{x}^{4}\log{x}}{4}-\dfrac{1}{4}\dfrac{{x}^{4}}{4}+c$$

$$\therefore\,I=\dfrac{{x}^{4}\log{x}}{4}-\dfrac{{x}^{4}}{16}+c$$
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Subjective Medium Published on 17th 09, 2020
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