Evaluate the integrals in Exercises 69–134. The integrals are ...

Question

Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.

127. ∫ (ln x) / (x + x ln x) dx

Accepted Answer

Welcome back everyone. Evaluate the integral ln x dx divided by x multiplied by 2 + ln x. For this problem we're going to use the u substitution. Let's suppose that u is equal to 2 + ln x, and let's remember that the next step is to find the derivative of u with respect to x. That's the derivative of 2 + ln of x, which is 0 + 1 divided by x. Notice that if we multiply both sides by dx, we can show that duu is equal to dx divided by x, and this is really useful because we have dx in the numerator and x in the denominator. Now, what we're going to do is also solve for Lon of X. If we subtract two from both sides of the original expression of U, that'd be U minus 2 equals eon of X, which allows us to rewrite the integral in terms of U and DU. So, a lot of X and the numerator becomes u minus 2. In the denominator we have 2 plus ellen of x which is u, and dx divided by x becomes duu. Well done. So now we can perform division. You divided by U is 1, and we are subtracting 2 divided by U. We're integrating with respect to you. And now we can split the integral into two parts. One of them is going to be integral of 1 DU, which is the integral of DU minus. Integral of 2 divided by uDU. We can also factor out the constant which is 2. And we can integrate one divided by you, the. So now we have two elementary integrals. The integral of DU is U. And we're subtracting 2 multiplied by the integral of d u divided by u, which is ln of the absolute value of u. Plus a constant of integration c. Well then, so now let's replace you with 2 + ln of x. We get 2 + ln of x minus 2 ln of the absolute value of. 2 plus ln of x plus a constant of integration c. Notice that we have an additional constant which is 2, so we can reassign the previous constant to c1, and c is going to be equal to c1 plus 2. Meaning we can just remove 2 and C will carry that 2 as well as c 1. So the final answer is ln of x minus 2 ln of the absolute value of 2 plus 11 of x plus c. Thank you for watching.

Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.127. ∫ (ln x) / (x + x ln x) dx

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Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.
127. ∫ (ln x) / (x + x ln x) dx