Problem 23

Question

Determine whether the improper integral diverges or converges. Evaluate the integral if it converges, and check your results with the results obtained by using the integration capabilities of a graphing utility. $$ \int_{0}^{1} \frac{1}{1-x} d x $$

Step-by-Step Solution

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Answer

The given improper integral diverges.

1Step 1: Identify Indefinite Integral

First identify the indefinite integral of the given integral. The integral of $\frac{1}{1-x}$ is $-\ln|1-x|$, and the definite integral becomes $-\ln|1-x|\Big|_0^1$

2Step 2: Evaluate Limits

Now evaluate the limits as $x$ approaches $0$ and $1$ from the left. $-\ln|1-1| + \ln|1-0|$ simplifies to $-\ln(0) + \ln(1)$. $\ln(1) = 0$, but $\ln(0)$ is undefined in real numbers which means the limit as $x$ approaches $1$ is infinite.

3Step 3: Determine Convergence or Divergence

Since the limit as $x$ approaches $1$ is infinite, the original integral diverges.

Problem 22

Problem 23

Other exercises in this chapter

Problem 22

Use partial fractions to find the indefinite integral. $$ \int \frac{1}{4 x^{2}-9} d x $$

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Problem 22

Find the indefinite integral. (Hint: Integration by parts is not required for all the integrals.) $$ \int x^{3} \ln x d x $$

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Problem 23

Approximate the integral using (a) the Trapezoidal Rule and (b) Simpson's Rule for the indicated value of $n$. (Round your answers to three significant digits

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Problem 23

Use partial fractions to find the indefinite integral. $$ \frac{5-x}{2 x^{2}+x-1} d x $$

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