9–40. Integration by parts Evaluate the following integrals us...

Question

9–40. Integration by parts Evaluate the following integrals using integration by parts.

38. ∫ x² ln²(x) dx

Accepted Answer

Hello there. Today we're to solve the following practice problem 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. Evaluate the integral, the integral of X to the power of 3 multiplied by the natural log to the power of 3 of XDX using integration by parts. Awesome. So it appears for this particular problem we're given this particular integral and we're asked to evaluate it using integration by parts. So now that we know what we're ultimately trying to solve for, our first step is we need to recall the integration by parts formula. So, as we should recall, the integration by parts formula states that the integral of UDV is equal to UV minus the integral of VDU. Awesome. So now that we've recalled the integration by parts formula, our next part of our part one, our first step that we need to take the next half of it, is we need to recall and use the acronym Lipit, which stands for L I P E T. And as you should recall, this acronym Lipi helps us choose the values of U and DV. So with that in mind, let us recall that L stands for a logamithic function such as the natural log of X. I stands for inverse trigonometrics, such as inverse tangent of X. P stands for polynomials such as X to the power of 2, or Y, etc. E stands for exponential, such as E to the power of X, and finally, T, our last letter in our acronym, stands for trigometric such as sign of X or Cosine of X, etc. Fantastic. So with that in mind, we need to note that paying attention to our original integral that is provided to us for the problem itself, we're told that the natural log to the power 3 of X, so that stands for logger mythic, and X to the power of 3 is a polynomial. So with that in mind, we will choose that. You will be equal to, so you will be equal to the natural log to the power 3 of X, which will mean that DU is equal to 3 multiplied by the natural log to the power 2 of X multiplied by 1 divided by XDX, fantastic. And DV will be equal to X to the power of 3D X, which will mean that V is equal to X to the power of 4 divided by 4. So now, we need to apply integration by parts. So when we apply integration by parts, we will note that we can write that the integral of X to the power of 3, multiplied by the natural log to the power of 3 of X D X is equal to. X to the power of 4 divided by 4 multiplied by the natural log to the power of 3 of x minus the integral of x to the power of 4 divided by 4 multiplied by 3 multiplied by the natural log to the power of 2 of x multiplied by 1 divided by XD X. And when we simplify, we will find that it's equal to x to the power of 4 divided by 4, multiplied by the natural log to the 3 X minus 3 divided by 4 multiplied by the integral of X to the power of 3, multiplied by the natural log to the power of 2 of XDX. Fantastic. So now we are at the point where we need to compute the interval of X to the power of 3 multiplied by the natural log to the power of 2 of XDX, and we need to repeat the integration by parts method. So with that in mind, our second step now is we need to apply integration by parts again. So this time we are focusing on the natural log. So we're focusing our attention on the following equation that has a natural log in it. So we're trying to apply integration by parts once again to this integral of X to the power of 3, multiplied by the natural log to the power 2 of X. DX and for this particular integral, which I've wrote in blue, we will determine that U is equal to the natural log to the power of 2 of X, which will mean that DU is equal to 2 multiplied by the natural log of X multiplied by 1 divided by XDX. And DV will be equal to X 3 DX, which will mean that V is equal to x 4 divided by 4, because once again DV is equal to x 3. OK. So, that will mean that we can go ahead and write that the integral of X 3 multiplied by the natural log to the power of 2 of X D X is equal to x 4 divided by 4, multiplied by the natural log to the power 2 of X minus the integral of X to the power of 4 divided by 4 multiplied by 2 multiplied by the natural log. Of X multiplied by 1 divided by X D X will be all equal to x 4 divided by 4 multiplied by the natural log to the 2 of X minus 1 divided by 2 multiplied by the interval of X 3 multiplied by the natural log of X D X. Fantastic. So now we need to compute the integral X to the power of 3, multiplied by the natural log of XDX. Fantastic. So, once again, We're going to be taking this new integral that we determined, and for a third step, we need to apply integration by parts again. So using once again we're paying attention to the integral of X to the power 3 multiplied by the natural log of XDX. So applying integration by parts to this particular integral, we will find that U is equal to the natural log of X, which will mean that DU is equal to 1 divided by X. DX and similarly, DV will be equal to X to the power of 3. DX which will mean that V is simply equal to x the power of 4 divided by 4. So that will mean that the integral of X 3 multiplied by the natural log of XDX will be equal to x 4 divided by 4 multiplied by the natural log of X minus the integral of X 4 divided by 4 multiplied by 1 divided by X. D X, which will be all equal to x 4 divided by 4 multiplied by the natural log of x minus 1 divided by 4 multiplied by the interval of x to the power of 3 D X which will be equal to x the power of 4 divided by 4, so X to the power of 4 divided by 4 multiplied by the natural log of x. Minus X to the power of 4 divided by 16. So moving right along here, our 4th step now is we need to plug back in our values step by step. So we need to note that at one point we had the integral. So once again we're plugging everything back into place. So we need to recall that we had the integral of X to the power 3 multiplied by the natural log to the power 2 of X. D X, which is equal to x 4 divided by 4, multiplied by the natural log to the 2 of X minus 12 multiplied by parenthesis X to the power of 4, so X to the power of 4 divided by 4 multiplied by the natural log of X minus x the power of 4. Divided by 16 and don't forget your parenthesis. So when we simplify, we will find that it's equal to x 4 divided by 4 multiplied by the natural log to the 2 of X minus parentheses X 4 divided by 8 multiplied by the natural log of x minus x 4 divided by 32. Which will be equal to x 4 divided by 4 multiplied by the natural log to the 2 of X minus X 4 divided by 8 multiplied by the natural log. Of Xs. Plus, X to the power of 4 divided by 32. Fantastic. So moving right along here, our fifth step now is we need to plug this back into our original expression, so we need to note that we have the integral of X to the power of 3, multiplied by the natural log to the power of 3 of X. DX will be equal to. X to the power of 4 divided by 4 multiplied by the natural log to the power of 3 of X minus 3 divided by 4 multiplied by the integral of X to the power of 3 multiplied by the natural log to the power 2 of X D X. So now, when we substitute in our values, we will find that it will be equal to X to the power of 4, divided by 4 multiplied by the natural log to the power of 3 X minus. 3 divided by 4 multiplied by parenthesis X to the power of 4 divided by 4 multiplied by the natural log to the power of 2 of X minus X to the power of 4 divided by 8 multiplied by the natural log of X plus x to the power of 4 divided by 32. And when we distribute, we will find that it's equal to x to the power of 4 divided by 4 multiplied by the natural log to the power of 3 of X minus 3 x 4 divided by 16 multiplied by the natural log to the power of 2 of X plus. 3 x to the power of 4 divided by 32 multiplied by the natural log of x minus 3 multiplied by x to the power of 4 divided by 128 and that is our final answer. Hooray, we did it. So, Looking at the problem itself, to recap what we've learned, that will mean that our final answer once again. Without a doubt, we'll have to be the following. So, with that in mind, our final answer is X to the power of 4 divided by 4, multiplied by the natural log to the power of 3 of X. Minus 3 multiplied by x to the power of 4 divided by 16 multiplied by the natural log to the 2 of X plus 3 multiplied by X to the power of 4 divided by 32 multiplied by the natural log of X minus 3 multiplied by X to the power of 4 divided by 128 plus C. and we can't forget this plus C because the C represents some constant. And that's it. That is our final answer. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

9–40. Integration by parts Evaluate the following integrals using integration by parts.
38. ∫ x² ln²(x) dx

Key Concepts

Integration by Parts

Logarithmic Functions and Their Derivatives

Polynomial Functions and Their Integration

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