Indefinite Integrals

Easymathematics

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

Select the correct option:

$e^{x} ln x + C$

$e^{x} / x + C$

$e^{x} x ln x + C$

$e^{x} (1 + ln x) + C$

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About This Question

Subject: mathematics
Chapter: integral calculus
Topic: indefinite integrals
Difficulty: Easy
Year: 2025

Solution

Correct Answer:

$e^{x} ln x + C$

Simplify the integrand: $\frac{1 + x l n x}{x} = \frac{1}{x} + ln x$ .
Recognize the standard form $\int e^{x} (f (x) + f^{'} (x)) d x = e^{x} f (x)$ . Let $f (x) = ln x$ . Then $f^{'} (x) = 1/ x$ .
The integral becomes: $\int e^{x} (ln x + \frac{1}{x}) d x = e^{x} ln x + C$ .

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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