I'm trying to do as many integration by parts problems as I can in order to get practice, but for some reason I keep getting stuck on some and just can't get anywhere with them (I move on to ones I can do though). For example here's a problem that I have the answer to but I just can't seem to imagine the steps that were taken to get the answer.
\(\displaystyle \int (t+2)\sqrt{2+3t}\) or \(\displaystyle \int (t+2)({2+3t}^{1/2})\)
My preferred pattern for integration by parts goes:
\(\displaystyle \int fg'=fg-\int f'g\)
So applying this to the problem I get:
\(\displaystyle \int (t+2)({2+3t}^{1/2})=(t+2)(\dfrac{2}{3}(2+3t)^{3/2}-\int (\dfrac{t^2}{2}+2t)(\dfrac{2}{3}(2+3t)^{3/2}\)
Chegg/Cramster and Wolfram are confusing to follow as well. I don't know, maybe I'm just in a rut with some of these.
\(\displaystyle \int (t+2)\sqrt{2+3t}\) or \(\displaystyle \int (t+2)({2+3t}^{1/2})\)
My preferred pattern for integration by parts goes:
\(\displaystyle \int fg'=fg-\int f'g\)
So applying this to the problem I get:
\(\displaystyle \int (t+2)({2+3t}^{1/2})=(t+2)(\dfrac{2}{3}(2+3t)^{3/2}-\int (\dfrac{t^2}{2}+2t)(\dfrac{2}{3}(2+3t)^{3/2}\)
Chegg/Cramster and Wolfram are confusing to follow as well. I don't know, maybe I'm just in a rut with some of these.