Indefinite Integrals

Standard Result: The power rule formula $\frac{x ^{n + 1}}{n + 1}$ fails for $n = - 1$ (division by zero).
Logarithmic Answer: The derivative of $ln (x)$ is $1/ x$ .
Absolute Value: Because $1/ x$ is defined for negative $x$ (where $ln x$ is not), we must include an absolute value to cover the entire domain of $1/ x$ .
Result: $\int \frac{1}{x} d x = ln ∣ x ∣ + C$ .

This easy difficulty mathematics question is from the chapter integral calculus, covering the topic of indefinite integrals. It appeared in the 2025 exam.

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Indefinite Integrals

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Solution

Evaluate $\int e^{x} (\frac{1 + x l n x}{x}) d x$ (Assume $x > 0$ and base $e$ for $ln$ ).