Question:
\(\displaystyle \int e^{x}\!\left(\frac{1}{x}-\frac{1}{x^{2}}\right)\,dx=\) ?
\(\displaystyle \int e^{x}\!\left(\frac{1}{x}-\frac{1}{x^{2}}\right)\,dx=\) ?
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A pattern like $e^x\!\left(\frac{1}{x}-\frac{1}{x^{2}}\right)$ screams "derivative of $\frac{e^x}{x}$".
Updated On: Oct 17, 2025
- \(\dfrac{e^{x}}{x}+k\)
- \(-x\,e^{x}+k\)
- \(k-\dfrac{e^{x}}{x}\)
- \(k-\dfrac{e^{x}}{x^{2}}\)
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The Correct Option is A
Solution and Explanation
Differentiate \(\dfrac{e^{x}}{x}\) using quotient rule:
\[
\left(\frac{e^{x}}{x}\right)'=\frac{x e^{x}-e^{x}}{x^{2}}=e^{x}\left(\frac{1}{x}-\frac{1}{x^{2}}\right).
\]
Matches the integrand, so the antiderivative is \(\dfrac{e^{x}}{x}+k\).
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