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

Question

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

11. ∫ t · e⁶ᵗ dt

Accepted Answer

Welcome back, everyone. Evaluate the integral of Y multiplied by E to the 3 YDY using integration by parts. For this problem, let's recall the integration by parts formula. The integral of UDB is equal to U minus the integral of VDU. For this problem, we have a product of two functions. Let's set you. Equal to Y. And then DV is going to be equal to the remaining part, which is E to the 3 Y D Y. We can now show that if u equals y, then D U is equal to Dy, right? And if we want to solve for V because we will need V. We can integrate E to the 3 YDY. The integral of E to the 3 Y is going to be E to the 3Y divided by 3. We're dividing by the slope of the linear term, right? So now we have the expression of V as well. And we are ready to apply the formula. So we can now show that the integral of Y multiplied by E to the 3 Y D Y. Is equal to U multiplied by V, so. Y multiplied by E to the 3 Y divided by 3. And we're subtracting the integral of VDU. So we're subtracting the integral of E to the 3 Y divided by 3. And the U is the Y. So now, let's focus on the final integral. We can factor out 1/3 because it's a constant, and we already know that the integral of E to the 3 Y is equal to. E to the 3 Y divided by 3. Because we already had this integration before. We integrated to solve for Y and it had the same integral, right? Now let's go ahead and simplify. We can show that our answer would be. First of all, why divided by 3. E to the 3 Y. Minus 1/3 Multiplied by e to the 3 Y divided by 3, and we want to add a constant of integration C because this is an indefinite integral. and at 1/3 multiplied by 1/3, there's 19th. So we can just add 1 9th in front. And perform factorization. So let's go ahead and factor it out. We can take out 19th. E to the 3 Y. The first term in parent becomes 3 because 3 divided by 9 is 1/3. Followed by why because we are also missing why. And we are subtracting one right because. We factored out our second term completely. We just want to subtract one to take into account that negative sign. And then we're adding a constant of integration C. Therefore, our final answer would be 1/9 E to the 3Y multiplied by 3Y minus 1 plus C. Thank you for watching.

9–40. Integration by parts Evaluate the following integrals using integration by parts.11. ∫ t · e⁶ᵗ dt

Watch next

9–40. Integration by parts Evaluate the following integrals using integration by parts.
11. ∫ t · e⁶ᵗ dt