Evaluate the integrals in Exercises 97–110.
103. ∫₁⁴ (ln...

Question

Evaluate the integrals in Exercises 97–110.

103. ∫₁⁴ (ln 2 · log₂x / x) dx

Accepted Answer

Welcome back everyone. Compute the integral from 1 to e cubed of 1 of 5 multiplied by a log of base 5 of x divided by xd x. AS 9 B 9 divided by 2, C3, and D 3 divided by 2. For this problem to make the problem simpler, let's change the base of the logarithm. We have log of base 5 of x, and we can change it to the natural logarithm. Let's remember that we take the natural logarithm of x and we divide by the natural logarithm of the original base, which is 5. So log of base 5 of x is eon of x divided by ln of 5. And now if we consider the function that we're integrating, we have LA of 5 multiplied by a log of base 5 of x, which is now Lon of x divided by Lon of 5. Divided by X. Notice that now we can cancel out LN of 5, and we are simply integrating LN of X divided by X, right? What we can do is use substitution. Let's suppose that U is equals to eon of x. The derivative of u with respect to x is the derivative of eon of x, and that's 1 divided by x. And we can multiply both sides by dx to show that dU is equal to dx divided by x. So now within our integral we have dx divided by x, which simplifies to duu. So the final step to use u substitution is to change the limits of integration. The lower 1, u1, is going to be ln of 1, which is 0, and the upper limit of integration u2 is ln of ecubed, which is 3. So the integral in terms of u and duu can be written as the integral from 0 to 3. Of now Een of x becomes you. And dx divided by x becomes DU. Now the integral of uDU is an elementary integral. We can use the power rule to find the value of this integral, and it is u 2 divided by 2, and we want to evaluate the result from 0 to 3 using the fundamental theorem of calculus part 2. We can factor out 1/2 and simply consider u squared from 0 to 3. So that's 32 minus 02, which is 9 minus 0, that's 9, and 9 multiplied by 1/2 is 9 halves. So the correct answer for this problem corresponds to the answer of choice B. Thank you for watching.

Evaluate the integrals in Exercises 97–110.103. ∫₁⁴ (ln 2 · log₂x / x) dx

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Evaluate the integrals in Exercises 97–110.
103. ∫₁⁴ (ln 2 · log₂x / x) dx