The integral

$I = \int_{0}^{1} \frac{0.5}{x-1} \, dx$

requires careful analysis because the integrand has a singularity at $x = 1$, making the integral improper. Let us address this step-by-step:

1. Rewrite the Integral

We simplify the constant factor: $I = 0.5 \int_{0}^{1} \frac{1}{x-1} \, dx$

2. Assess the Singularity

The term $\frac{1}{x-1}$ becomes undefined as $x \to 1$. Therefore, the integral is improper, and we split it into two parts, approaching the upper limit $x = 1$ as a limit: $I = 0.5 \lim_{b \to 1^-} \int_{0}^{b} \frac{1}{x-1} \, dx$

3. Compute the Antiderivative

The antiderivative of $\frac{1}{x-1}$ is: $\int \frac{1}{x-1} \, dx = \ln|x-1| + C$

Thus: $\int_{0}^{b} \frac{1}{x-1} \, dx = \ln|b-1| - \ln|0-1| = \ln|b-1| - \ln(1)$

4. Evaluate the Limit

Substitute back into the integral: $I = 0.5 \lim_{b \to 1^-} \ln|b-1|$

As $b \to 1^-$, $\ln|b-1| \to -\infty$. Therefore, the integral diverges to $-\infty$.

Conclusion

The integral:

$\int_{0}^{1} \frac{0.5}{x-1} \, dx$

does not converge. The result is $-\infty$.

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Would you like me to explore this concept further or analyze a similar problem?

Related Questions:

1. How do improper integrals with logarithmic singularities behave in general?
2. Can this integral be computed on different intervals (e.g., $[a, b]$ avoiding $x=1$)?
3. What are the practical applications of singular integrals in physics or engineering?
4. How can one numerically approximate improper integrals with divergence?
5. What is the significance of the natural logarithm in evaluating indefinite integrals?

Tip:

Always check the domain and behavior of the function before attempting to integrate, especially near singularities!