Mathematics

Write a value of 
$$\displaystyle\int { { e }^{ x }\left( \cfrac { 1 }{ x } -\cfrac { 1 }{ { x }^{ 2 } }  \right)  } dx$$


SOLUTION
$$I=\displaystyle\int{{e}^{x}\left(\dfrac{1}{x}-\dfrac{1}{{x}^{2}}\right)dx}$$

$$=\displaystyle\int{{e}^{x}\dfrac{1}{x}dx}-\displaystyle\int{{e}^{x}\dfrac{1}{{x}^{2}}dx}$$

Let $$u={e}^{x}\Rightarrow\,du={e}^{x}dx$$

$$dv=\dfrac{1}{{x}^{2}}dx\Rightarrow\,v=\dfrac{-1}{x}$$

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

Integrating by parts, we get

$$=\displaystyle\int{{e}^{x}\dfrac{1}{x}dx}+\dfrac{1}{x}{e}^{x}-\displaystyle\int{{e}^{x}\dfrac{1}{x}dx}+c$$

$$=\dfrac{1}{x}{e}^{x}+c$$
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Subjective Medium Published on 17th 09, 2020
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