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Definite Integral of 1/x * x/(1-x) from 0 to 1
Published on August 27, 2024
Calculus
Learn how to solve the definite integral of 1/x * (x/(1-x)) from 0 to 1. This step-by-step guide demonstrates simplifying the integrand, applying integration rules, and evaluating the limits to reveal the integral diverges to positive infinity.
Question
\int_0^1 \frac{1}{x} \left(\frac{x}{1-x}\right) dx
Answer
Let’s solve this integral step by step:
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First, let’s simplify the integrand:
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Now our integral becomes:
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This is a standard integral form. We know that:
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Now let’s apply the limits:
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Evaluating at the upper limit (x = 1):
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Evaluating at the lower limit (x = 0):
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Therefore, the result is:
Thus, the integral diverges to positive infinity.
Final answer: The integral