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

Question

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

∫ (x² - 2x + 1) e^(2x) dx

Accepted Answer

Welcome back, everyone. In this problem, we want to compute the integral of x2 minus 4 x + 3 all multiplied by e x with respect to x using integration by parts. Now what do we know about integration by parts that will help us to solve this problem? Well, recall that in integration by parts, we're basically saying that the integral of UDV is equal to UV minus the integral of VDU. So if we can set one of these terms to be you and the other to be DV, then we can substitute into the integration by parts formula to solve. But now, the question is which one of them do we set to be you and which one is DV. Well, we can use the acronym LET to help us. It's commonly referred to as the Lipe rule, and basically it helps us to prioritize which type of function we would pick. Logs would be first, then inverse trigonometric functions, then polynomials, then exponentials, then trigonometric functions. Now in this case, notice here that we have a polynomial x2 minus 4 x + 3. And then we have an exponential function that is e x. So based on the lipid hierarchy where polynomials have a higher priority, then let's set u to be equal to x2 minus 4 x + 3, OK. And dV to be equal to e xDX. Now in this case we can find a derivative of you with respect to x which would be equal to 2 x minus 4, OK, which means that duu would be equal to 2 x minus 4 multiplied by dx. On the other hand, if DV equals e xDX, if we integrate both sides, then we get V to be equal to e x. So no applying the integration by parts formula, that means the integral of x2 minus 4 x + 3 multiplied by e x with respect to x would be equal to uv x2 minus 4 x + 3 multiplied by e x. Minus the integral of VDU, so that's e to the x multiplied by 2 x minus 4 d x. I know if we start to simplify here, OK, then we end up with X2 minus 4 X + 3 all multiplied by e to X, OK, minus 2 multiplied by the integral of xe to XDX, OK, plus 4 multiplied by the integral of e to the XDX. I know if we integrate what we can, OK, then we're going to end up with X2 minus 4 X + 3. Multiplied by e x plus 4 e x because the integral of e to the x with respect to x is e to the x minus 2 multiplied by the integral of x e to the x d x. Since both of these terms have eat the eggs. Then we can combine them to get X2 minus 4 x + 3 x + 4 E X R + 7 factored out e to the x. So now we're almost there, but if you notice again, we'll need to figure out the integral of x e to the x with respect to x. How can we do that? Well, again, we can use integration by parts. So in this case, OK. Integrating. Xe to the x with respect to the x by parts. Again, following the same hierarchy as using polynomials first or choosing a polynomial to be u over an exponential, OK, then in this case we can let u be equal to x, OK. And we'll let DV. Be equal to e x dx. In that case, the derivative of u with respect to x is 1, so d u equals d x, and integrating dV equals e to the x dx, we get v to be equal to e to the x. So now if we apply the integration by parts formula in this case, then the integral of xe to the xDx. Will be equal to. X e to the x minus the integral of e to the xd x, but now we know what that integral is, so this would simplify to x x multiplied by e x minus e to the x. And now we can substitute that back into our original result or equation, OK. So now when we do that. OK. Then, going back to our original problem of the integral of X2 minus 4 x + 3. OK, the multiplied by e to the XDX. Let me make some space here. Then it's going to become x2 minus 4 x + 7 multiplied by e to the x. -2 multiplied by xe to x minus e x, OK, plus c. And now again, here's an opportunity to factor out e to the x, so that's going to be X2 minus 4 x + 7 minus 2 x + 2, all multiplied by e to the x. And when we simplify here, that becomes x2 minus 6 x + 9 multiplied by e x + c. So this would be the integral of x2 minus 4 x + 3 all multiplied by e x with respect to x if we were to use integration by parts. Thanks a lot for watching everyone. I hope this video helped.

Evaluate the integrals in Exercises 1–24 using integration by parts.∫ (x² - 2x + 1) e^(2x) dx

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Evaluate the integrals in Exercises 1–24 using integration by parts.
∫ (x² - 2x + 1) e^(2x) dx