The integrals in Exercises 1–34 converge. Evaluate the integra...

Question

The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.

∫₀¹ (−ln(x)) dx

Accepted Answer

Welcome back everyone. Evaluate the improper integral from 0 to 5 of E of XDX. For this problem, let's go ahead and rewrite the integral, integral from 0 to 5 of LN of X D X. Now, a lot of X is undefined at 0, so what we're going to do is use the limit. Limit of the integral, we're going to change 0 to some letters such as b. So from B to 5, as B approaches 0 of LN of XDX. This is going to be limit as b approaches 0. And now for the integral, we're going to use the result from integration by parts. So we can just write the result as follows the integral of L of x. D X is going to be x. Alan of x minus x. That's our integral, and we want to evaluate it from B up to 5, right? Additionally, B in our limit is going to approach 0 from the right because essentially if we consider the graph of the natural logarithm. The limit exists from the right. And now let's go ahead and apply the limits of integration, which gives us limit as B approaches 0 from the right. Of 5 LN of 5 minus 5 minus. In parentheses. B Ellen of B. Minus B. Now let's simplify this limit S. B approaches 0 from the right. Of 5 ln of 5 minus 5 + b ln of b + b. Now, let's use the properties of limits. The limit of a constant is the constant itself, so we get 5 ln of 5 minus 5. Plus, Limits of BSB approaches. 0 from the right is 0, right, and we're left with limit as b approaches 0 from the right of B ln of B. If we consider this limit, we're going to get 0 multiplied by eon of 0, which is an indeterminate form 0 multiplied by. Negative infinity. So instead what we can do is rewrite it as ln of b divided by 1 minus b because it will give us an indeterminate form. For which we can apply Lapto's rule now now we will have. 1 divided by 0 from the right, which is going to be. A limit of infinity and LN of B is going to give us a limit of negative infinity, so this allows us to apply Lato's rule. So we now have 5 L of 5 minus 5 plus. Limit as b approaches 0 from the right applying Lapito's rule. The derivative of Eon of B is 1 divided by B. And the derivative of 1 divided by b is -1 divided by b 2. Now let's simplify 5 of 5 minus 5 plus. Limit. As B approaches 0 from the right. Of So from here we have 1 divided by B divided by negative, 1 divided by B2, and that's basically g B. Well done, and now SB approaches 0 from the from the right. Limit of negative b is going to be zero. We can just substitute, so we can just cross out that term because it is equal to 0, and this is how we get our final answer, which is. 5 EN of 5. -5. And that's our final answer. Let's label it and thank you for watching.

The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.∫₀¹ (−ln(x)) dx

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The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₀¹ (−ln(x)) dx