Evaluate the integrals in Exercises 1–24 using integration by ...

Question

Evaluate the integrals in Exercises 1–24 using integration by parts.

∫ t² e^(4t) dt

Accepted Answer

Welcome back everyone. In this problem we want to compute the integral of X2E2X with respect to X using integration by parts. Now what do we know about integration by parts that will help us? Well, recall that it basically says that if we're finding the integral of UDV, it will be equal to UV minus the integral of VDU. So if we assign you and DV to each of these terms in our integral, then we should be able to differentiate or integrate to be able to solve for our integration by parts. So let's let you be equal to x2, OK. And let's let DV be equal to E 2 X, OK, with respect to x. Well, if U equals x2, that means the derivative of u with respect to x would be equal to 2x, which means that duu will be equal to 2 x dx. On the other hand, if dV equals e to the 2 x dx, if we integrate both sides, the integral of dv will be equal to v, and the integral of e to the 2 x with respect to x will be equal to a half of e to the 2 x. So now that we have v and DU, we can apply our integration by parts because this means then that the integral of x 2 e to the 2 x with respect to x, OK, substituting in our formula will be equal to uV, that's x2 multiplied by a half of e to the 2 x. Minus the integral of vDu, that's a half of e to the 2 x multiplied by 2 x d x. And now if we simplify here, OK, we should get this to be a half of x 2 e to the 2 x, OK, minus the integral of xe to 2 x with respect to x because 12 multiplied by 2 equals 1 within our integral. So now we're on our way, but if you notice here, we need to integrate again. We need to integrate Xe to the 2 X with respect to x. How can we do that? Well, again, we can apply integration by parts. So integrating. OK, integrating our expression, that is Xe to the 2 X with respect to X by parts, OK. In this case, we can let X, or sorry, let you. Be equal to x. And let's let DV. Be equal to e to the 2 x with respect to x. Now again, if u equals x, d u, the derivative of u with respect to x will be 1, so duu will be equal to dx. And if dV equals e to the 2 x with respect to x, if we integrate both sides, then v will be equal to a half of e to the 2 x. So know that we have this. This means then that our integral of x e to the 2 x with respect to x will be equal to uv x multiplied by a half of e to the 2 x minus the integral of VDU. So that's minus the integral of a half e to the 2 x with respect to x. Now if we simplify here, this will be equal to. A half of x e to the 2 x minus a half of the integral of e to the 2 x with respect to x, and we know that the integral of e to the 2 x with respect to x will be equal to, let me just write this here. So this is going to be a half of E to the 2 X. I know a 1/2 multiplied by 1/2 of e to the 2 x, OK, will eventually simplify to 25 of e 2 x. So this will be a, well, let me just write it here, this will be a half of xe to 2 x minus minus a quarter. -25 of e to the 2 X. And now that we have an expression for this, we can put it back into our expression here. So if that's the case, if we go back to it, that means our integral, OK, of X. S2 multiplied by e to 2 x with respect to x will be equal to 12 of x 2 multiplied by e 2 x minus this integral that we just found, which is 122 of x e to the 2 x minus a quarter of e to the 2 x plus c. I know when we simplify here, OK. Then this will be, we'll get a 12 of x2e to the 2X minus 12 of XE 2 X plus a quarter of e to the 2 x plus C. Which will be equal to if we factor out we'll have a quarter of e to the 2 x multiplied by 2 x 2 minus 2 x + 1 + c. So this is the integral of x2 multiplied by e 2 x with respect to x integrating by parts. Thanks a lot for watching everyone. I hope this video helped.

Evaluate the integrals in Exercises 1–24 using integration by parts.∫ t² e^(4t) dt

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Evaluate the integrals in Exercises 1–24 using integration by parts.
∫ t² e^(4t) dt