Evaluate the integrals in Exercises 39–56.
45. ∫(from 1 ...

Question

Evaluate the integrals in Exercises 39–56.

45. ∫(from 1 to 2)(2ln x)/x dx

Accepted Answer

Welcome back, everyone. Evaluate the D integral from 1 to 4 of 3 LN X D X divided by X. A says 3 LN of 4, B 3 halves multiplied by LN2 of 4, C 6LN of 4, and D 34s multiplied by LN2d of 4. For this problem, let's go ahead and use the U substitution. Let's suppose that U is equal to LN X. We want to identify the derivative of you with respect to X, which is a 1 divided by X. If we multiply both sides by the X, we can show that the U is equal to D X divided by X, which we have, right, in the integras D X divided by X. We also want to identify the new limits of integration. The lower 1 U1 is LN of 1, which is equal to 0, and the upper one is LN of 4. So now we can rewrite our integral in terms of UNDU. We have integral from 0 up to LN of 4 of we multiplied by LN of X, which is U, multiplied by D X divided by X, which is D U. We can factor out the constant which is 3, and integrate UDU from 0 up to LN of 4. This is an elementary integral, we can apply the power rule to get we multiplied by U 2 divided by 2. And we want to evaluate the result from 0 up to LN04 using the evaluation theorem. We can factor out 3 halves, and in C we can focus on U squared. First of all, let's substitute LN of 4 to get LN squad of 4 minus. You squared when you 0 is 0 squared. So now let's go ahead and simplify that 3 halves multiplied by LN squared of 4. We can also write it in a different style, that'll be 3 halves multiplied by LN of 4. Squared rights and this corresponds to the answer choice b. Thank you for watching.

Evaluate the integrals in Exercises 39–56.45. ∫(from 1 to 2)(2ln x)/x dx

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Evaluate the integrals in Exercises 39–56.
45. ∫(from 1 to 2)(2ln x)/x dx