Mathematics

Evaluate: $$\displaystyle \int{e^x}\begin{pmatrix}\dfrac{x-1}{x^2}\end{pmatrix}dx$$.


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

Let $$\dfrac { 1 }{ x } =f(x),f'(x)=\dfrac { -1 }{ { x }^{ 2 } } $$
$$I=\int { { e }^{ x }\left( \frac { 1 }{ x } -\dfrac { 1 }{ { x }^{ 2 } }  \right)  } $$

Using the property, we get:
$${ e }^{ x }(f(x)+f'(x))dx={ e }^{ x }f(x)+c$$
$$I=\dfrac { { e }^{ x } }{ x } +c$$
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Subjective Medium Published on 17th 09, 2020
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