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

Question

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

32. ∫ from 0 to 1 x² 2ˣ dx

Accepted Answer

Evaluate the integral from 0 to 1 of 5 X X 2DX using integration by parts. Now, I've written a little chart here on the right to help us select our use substitution. But first, we know the integration by parts formula is given by the integral of UDV equals UV minus the integral of VDU. Now this chart on the right will help us select the U. In this case, we have 5 to the X, which is an exponential. An x squared which is a polynomial. Now because P is polynomial, and that comes before the exponential, we'll say U equals X squared. DU then will be 2 X. And DV will be 5 races to the X. V is the interval of 5 to X, which is just 5 X divided by natural log of 5. Now we can go ahead and fill out our integration by parts. We have U, which is X squared multiplied. By the 5 versus the X divided by natural log of 5. Minus the integral. Uh 5 x divided by national log of 5. DU, which is 2 X. Now we can take another integral. Now we noticed that this is another integration by parts. So we'll go ahead and leave our first term here and just focus on the second term. We have the integral to the X divided by natural log of 5. Multiplied by 2 XDX. You will once again be 2 X. Which makes DU to DX. DV will be 5 to the X divided by the natural log of 5. And V will be the antiderivative, which is 5 X divided by natural log of 5 squared. Now we can plug this into our new integration by parts. We have 2 X multiplied. By the 5 to the X divided by natural log of 5 square. Minus the integral of V 5 to the X divided by natural log of 5 squared multiplied. By DU, which is 2DX. Now we can find this last integral. We have 2 X multiplied. By 5 to the X divided by natural log of 5, all squared. -2 multiplied by 5 x divided by natural log of 5. Race to the 3rd. Now we can combine all of our terms into one final integral. We have X to the 2nd, multiplied by 5 X divided by natural log of 5. Minus in parentheses. We have 2 X multiplied by 5 to the X divided by natural log of 5 all squared. -2 multiplied by 5 to the X, divided by natural log of 5, all cuts. We also have bounds from 0 to 1. No. Let's go ahead and plug in our terms. We end up getting one squared multiplied. By 5 races to the first divided by natural log of 5. -2 multiplied by 1. 5 to the first divided by natural log of 5 square. -2 multiplied by 5, divided by natural log of 5, raise to the third. And then the same, but with 0. We have Minus 0 squared multiplied by 5 to 0, divided by the natural log of 5. Minus 2 multiplied by 0. Multiply by 5 to 0. Developed by Nala of 5 squid. -2 multiplied by 5 to 0, divided by natural log of 5, raise to the third. We can finally simplify, to get 5 divided by natural log of 5. Minus 10 divided by natural log of 52. Minus 10 divided by natural log of 5 cuts. Plus 2 divided by natural log of 5 cuts. Finally we can simplify. To get 5 divided by natural log of 5. -10 divided by natural log of 5 square. 8 divided by natural log of 5 cubs. This is the final answer to our problem. OK, I hope to help you solve the problem. Thank you for watching. Goodbye.

9–40. Integration by parts Evaluate the following integrals using integration by parts.32. ∫ from 0 to 1 x² 2ˣ dx

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9–40. Integration by parts Evaluate the following integrals using integration by parts.
32. ∫ from 0 to 1 x² 2ˣ dx