Evaluate the integrals in Exercises 39–56.
47. ∫(from 2 ...

Question

Evaluate the integrals in Exercises 39–56.

47. ∫(from 2 to 4)dx/(x(ln x)²)

Accepted Answer

Welcome back, everyone. Evaluate the D integral from 3 to 9 of the X divided by X, which is multiplied by LN of X squared. A 1 divided by 2 LN 3 B 1/4 LN of 3, C 2 LN of 3, and D 3 divided by LN of 2. For this problem, let's go ahead and use U substitution. Let's suppose that U is equal to LN X. Then we can show that the derivative of U is equal to 1 divided by X. Multiplying both sides by the X, we can now show that D X divided by X is equal to the. You, right? So now from here. We can rewrite our integral in terms of you and the U, but before we proceed, let's identify the new limits of integration. The lower limit of integration is going to be U1 equals 1 and 3, and the upper limit of integration is going to be LN 9. We're simply replacing X with 3 and 9 respectively. So now the integral becomes integral from. Alan of 3, up to Alan of 9. Notice that in the numerator we have the X in the nominator, we have X. So DX divided by X gives us D U. And we're dividing by LN of X2, which is U2. Now we have an elementary integral, we can rewrite it as U to the power of negative 2. The EU from Len of 3 up to LN of 9. And we can evaluate it using the power rule. So we get you to the power of -1 divided by -1, or basically negative U to the power of -1, which can also be written as -1 divided by U. Evaluated from LN of 3 up to LN of 9. And from here, we're going to apply the fundamental theorem of calculus part 2. So we get -1 divided by LN of 9. Minus -1 divided by ln of 3. Simplifying, we get positive 1 divided by LN of 3, minus 1 divided by LN of 9. What we can do is apply the properties of logarithms because we can rewrite 9 as we squared, and we can bring down the exponent, right, which basically gives us 1 divided by LN of 3 minus 1 divided by 2 LN of 3. And now we can find the common denominator, right, which is 2 LN of 3, multiplying the first fraction by 2 and dividing by 2, we get 2. In the numerator and we're subtracting one. So 2 minus 1 gives us 1, and this is the final answer. 1 divided by 2 LN of 3. Which corresponds to the answer choice A. Thank you for watching.

Evaluate the integrals in Exercises 39–56.47. ∫(from 2 to 4)dx/(x(ln x)²)

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Evaluate the integrals in Exercises 39–56.
47. ∫(from 2 to 4)dx/(x(ln x)²)