In Exercises 39–48, use an appropriate substitution and then a...

Question

In Exercises 39–48, use an appropriate substitution and then a trigonometric substitution to evaluate the integrals.

∫ √(1 - (ln x)²) / (x ln x) dx

Accepted Answer

Hello there. Today we are going to solve the following practice from together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Find the interval, the interval of the square root of 9 minus parenthesis, the natural log of x to the power of 2, all divided by X multiplied by the natural log of X V X. OK. So it appears for this particular problem we're asked to find the interval that is given to us by the problem itself. So we need to compute the interval. We need to solve for the interval. So now that we know what we're ultimately trying to solve for, let us note once again that we are told that the integral. Of the square root of 9 minus parenthesis, the natural log of x to the power of 2 all divided by X multiplied by the natural log of XB X. So this is the interval that is given to us by the prom itself and using this interval, let us state that you. will be equal to the natural log of X. That will mean then that DU by D X will be equal to 1 divided by X. So that will mean that we can go ahead and write that DU will be equal to 1 divided by XD X. Fantastic. So now what we need to do is we need to substitute back into our integrals. So we need to substitute our new values into our integral. So we can go ahead and write that the interval of the square root of 9 minus parenthesis, the natural log of x to the power of 2, all divided by X multiplied by the natural log of X D X is equal to the interval of. The square root of 9 minus u to the 2 all divided by U D U. fantastic. So our next step now is we need to recall and use trigontric substitution to state that U is equal to 3 multiplied by sine of u. So once again, so I, I need to say that more clearly. So U is equal to 3 multiplied by sine of theta. That will mean then So if U is equal to 3 multiplied by sin of theta using trigonometric substitution, then that will mean, without a doubt that DU by D theta will be equal to 3 multiplied by cosine of theta, which will mean that DU is equal to 3 multiplied by cosine of theta D theta. Also, we need to note that the square root of 9 minus u to the power of 2 will be equal to the square root of 9 minus parenthesis 3 multiplied by sin of beta to the power of 2, which will be equal to the square root of 9 minus 9 multiplied by sin to the power of 2 of theta. Which will be equal to the square root of 9 so it's going to be multiplied by parenthesis of 1 minus signed to the power of 2 of theta, which will mean it will be all equal to the square root of 9 multiplied by. Cosine square to theta, which will be equal to when we simplify one last time, 3 multiplied by cosine of theta. Fantastic. So now what we need to do is we need to substitute back into our integral. So we need to take the integral of the square root of 9 minus u to the 2 divided by U D U is equal to the. Integral of 3 multiplied by cosine of theta, divided by 3 multiplied by sine of theta, multiplied by 3 multiplied by cosine of theta. The theater. Fantastic. And this will be equal to, so drum roll, this is equal to 3 multiplied by the interval of cosine squared of theta, divided by sin of theta D theta, which will be equal to 3 multiplied by the interval of parentheses, 1 minus sin to the power of 2 of theta, divided by sine of theta D theta. Which will be equal to 3 multiplied by the integral of parentheses, 1 divided by sin theta minus square of theta, divided by sin of theta. The theta. Which will be all equal to 3 multiplied by the integral of parenthesis. Poicant of theta, so it's going to be equal to 3 multiplied by the interval of parenthesis. Posicant of theta minus sine of theta multiplied by D theta. Fantastic. Moving right along here, we also need to recall that the integral of Cosicant of theta D theta is equal to negative, the natural log of the absolute value of Cosicana theta plus cotangenta theta, once again, in absolute value signs plus C. And let us take a moment to note that any value inside of absolute value signs will become a positive value and that C represents some constant value. We also need to recall that, so and we need to recall that the integral of sine of theta D theta will be equal to negative cosine of theta plus C. So, therefore, we can go ahead and write that 3 multiplied by the integral of parentheses cosicant of theta minus, so it's going to be 3 multiplied by the integral of parentheses cosicana theta minus sin theta. D theta is equal to 3 multiplied by parenthesis negative the natural log of the absolute value of cosicant of theta plus cotangent of theta. Plus cosine of theta plus C. Which will be equal to -3 multiplied by the natural log of the absolute value of cosicant of theta plus cotangent of theta plus 3 multiplied by cosine theta plus C. Fantastic. So now we need to backs substitute sin of theta, so we need to backs substitute sin theta is equal to you divided by 3, to obtain that cosicanta theta is equal to 3 divided by you. So that will mean that Cosine theta is equal to the square root of 9 minus u2 divided by 3, and that cotangent of theta will be equal to the square root of 9 minus u to power 2 divided by u. So with that in mind, we can go ahead and write that -3 multiplied by the natural log of the absolute value of 3 divided by U plus the square root of 9 minus U 2 divided by you. Plus 3 Multiplied by the square root of 9 minus U 2 divided by 3 plus C. Will be equal to -3 multiplied by the natural log of the absolute value of 3 plus the square root of 9 minus u to power 2 all divided by u plus the square root of 9 minus u to the power of 2 plus C. So now we need to note that we need to replace U is equal to the natural log of X, and when we replace U to equal the natural log of X, we will be able to note that our final answer without a doubt will be. 3 multiplied by the natural log of the absolute value of 3 plus the square root of 9 minus parentheses the natural log of x to the power of 2 all divided by the natural log of X plus the square root of 9 minus parentheses the natural log of x to the power of 2. Plus C. And this is our correct and final answer. Hooray, we did it. So going back to look at our problem itself and to recap everything that we've learned, we will note that our final answer for the integral without a doubt, will be as follows. So our final answer once again is -3 multiplied by the natural log of the absolute value of. 3 plus the square root of 9 minus parentheses the natural log of x to the power 2 all divided by the natural log of x plus the square root of 9 minus the natural log of x in parentheses to the power 2 plus c. Fantastic, we did it. And that's it. Thank you so much for watching. Hopefully that helped and I can't wait to see you in the next video. Bye.

In Exercises 39–48, use an appropriate substitution and then a trigonometric substitution to evaluate the integrals.∫ √(1 - (ln x)²) / (x ln x) dx

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In Exercises 39–48, use an appropriate substitution and then a trigonometric substitution to evaluate the integrals.
∫ √(1 - (ln x)²) / (x ln x) dx