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

Question

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

29. ∫ e⁻ˣ sin(4x) dx

Accepted Answer

Hello there. Today we're 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. Compute the integral of E to the power of negative X multiplied by sin of 2XDX using integration by parts. Awesome. So it appears for this particular problem we're given this specific integral and we're asked to compute it using integration by parts. So now that we know what we're trying to solve for, our first step that we need to take in order to solve this problem is we need to recall and use Lipi, which is an acronym that stands for LIP. ET and this acronymII will help us choose our values for you and DV. So let's quickly recall what lipid stands for. So, L stands for logamithic functions, such as the natural log of X. I stands for inverse trigonometrics, such as Inverse tangent of X. P stands for polynomial, such as X to the power 2 or Y. E stands for exponential, such as E to the power of X, and finally T stands for trigontric such as sine of X or cosine of X, etc. So, with that in mind, let us choose our values for you and DV. So let us note that we need to recognize that looking at our integral that is provided to us, E to the power of negative X is an exponential. So this will be rank E. And sin of 2 X will be trigontric, so that will be rank T. So according to lipid, as we should recall and note, the exponential comes before trigontric. So we will choose that U is equal to E to the power of negative X. And DV will be equal to sin of 2 XDV so it's gonna be DV is gonna be equal to sin of 2 XDX. Fantastic. So now we need to compute, so we need to compute that DU is equal to negative E to the power of negative XDX fantastic. And we need to note that V is equal to the interval of sine of 2 XDX, which is equal to negative 12 or 1 divided by 2 multiplied by cosine of 2 X. Fantastic. Our second step now is we need to apply the integration by parts. So we need to recall that the interval of UDV is equal to UV minus the integral of VDU. So with that in mind, we can go ahead and write that the integral of E to the power of negative X. Multiplied by sin of 2 X D X is equal toe to the power of negative X multiplied by parenthesis negative 1/2 multiplied by cosine of, so negative 1/2 multiplied by cosine of drum roll 2 X. Don't forget your parentheses, minus the integral of negative 1/2 multiplied by cosine of 2 X multiplied by parenthesis, negative E to the power of negative XDX, fantastic. So it looks like we've run out of room here, so we'll shift this over just a little bit. There we go. So, We will note that this will be all equal to, so moving around here, this will be all equal to -12 multiplied by e to the power of negative X multiplied by cosine of 2X minus 1/2 multiplied by the interval of E to the power of negative X multiplied by cosine of 2 XD X. Fantastic. We are on our roll. So our third step now is we need to integrate the integral of E to the power of negative X multiplied by cosine of 2 XDX. fantastic. So once again, we need to use integration by parts again with. So we using, once again, we're using integration by parts with. Once again, we need to note that for this particular case now, U is going to be equal toe to the power of negative X, and DU will be equal to negative E to the power of negative XDX, and DV will be equal to cosine of 2 XDX. And drum roll and V will be equal to 12 multiplied by sin of 2 X. So that will mean therefore, that the integral of E to the power of negative X multiplied by cosine of 2 XDX. So once again, we'll be. So once again, we're taking the integral of E to the power of negative X cosine of 2 XDX will be equal to E the power of negative X multiplied by 1/2 multiplied by sin of 2 X minus the integral of 1/2 multiplied by sin of 2 X multiplied by parenthesis, negative to the power of negative X. DX fantastic. And this will be all equal to -12 multiplied by E to the power of negative X multiplied by sin of 2 X plus 1/2 multiplied by the integral of E to the power of negative X multiplied by sin of 2XDX. And we now need to put it all together for our 4th step, and we need to recall from earlier that the integral of E to the power of negative X multiplied by sin of 2 X D X is equal to -12 multiplied by E to the power of negative X multiplied by cosine of 2X minus 12. Multiplied by the integral of E to the power of negative X multiplied by cosine of 2 XDX. Fantastic. And we now need to substitute this result from step 3. So we're going to be substituting in our results from step 3, and when we do that, we will find that we will be able to write the following. So we'll be writing that this is all equal to -12. Multiplied by the power of negative X multiplied by cosine of 2 X minus 12 multiplied by parenthesis, 1/2 multiplied by E to the power of negative X multiplied by sin of 2 X plus 1/2 multiplied by, so. We have sin of 2 X plus 1/2 multiplied by the integral of E to the power of negative X multiplied by sin of 2 XDX, fantastic, we are on a roll here. And we now need to simplify, and when we simplify, we will find that it's simply equal to -12 multiplied by e to the power of negative X multiplied by cosine of 2X minus 14th multiplied by E to the power of negative X multiplied by sin of 2X minus 1/4. Multiplied by the integral of E to the power of negative X multiplied by sin of 2 XD X. So now our fifth step is we need to solve algebraically. So we need to let I equal the integral of E to the power of negative X multiplied by sin of 2 XDX. Then we could state that I is equal to -12 multiplied by E to the power of negative X multiplied by cosine of 2 X. Minus 1/4 multiplied by E to the power of negative X multiplied by sin of 2X minus 1/4 multiplied by I. So we need to bring 14th I to the left hand side, and when we bring it to the left hand side, we will find that it will be I plus 1/4 I is equal to 1/2 multiplied by the power of negative X multiplied by cosine of 2X minus 14 multiplied by E to the power of negative X multiplied by sin of 2 X. Fantastic. And once again, we're trying to simplify. And when we do, when we start to simplify, we will get 5 divided by 4. I is equal to -12 multiplied by E to the power of negative X multiplied by cosine of 2 X minus 1/4 multiplied by E to the power of negative X multiplied by sin of 2 X. And when we multiply both sides by 4 divided by 5, we will find that i is equal to negative 2 divided by 5 multiplied by e to the power of negative X multiplied by cosine of 2X minus 15th multiplied bye the power of negative X multiplied by sin of 2 X plus C. And when we simplify one last time, we will find that this will be equal to -15 multiplied by E to the power of negative X multiplied by parenthesis 2 multiplied by cosine of 2 X plus of 2 X in parenthesis plus C. and that's it. That is our final answer. Hooray, we did it. So to quickly recap what we've learned, once again, our final answer without a doubt will have to be the following. So looking at the prom itself. We will find that our final answer once again will have to be negative 1/5 multiplied by e to the power of negative X multiplied by parenthesis, 2 cosine of 2 X plus sine of 2 X plus C. 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.
29. ∫ e⁻ˣ sin(4x) dx

Key Concepts

Integration by Parts

Integration of Exponential and Trigonometric Functions

Solving for the Original Integral

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