Can anybody prove that the following equation is right or wrong? $$\int_0^te^{-t}(1-e^{-2x})^ke^x dx=\int_0^t2k(e^{-2x}-e^{-t-x})(1-e^{-2x})^{k-1}dx$$
where $t>0$ and $k$ is and integer. My small numerical evaluation showed that this equation is right.(e.g. I calculated the left and right hand side for t=.1 and k=3). Any help in this regard will be highly appreciated.
BR
Frank
Applying integration by parts one has $$\int_0^te^{-t}(1-e^{-2x})^ke^x dx=\left(1-e^{-2t}\right)^k-\int_0^t2ke^{-t-x}(1-e^{-2x})^{k-1}dx.$$ Also, one may note that $$\frac{d}{dx}\left(1-e^{-2x}\right)^k =2ke^{-2x}\left(1-e^{-2x}\right)^{k-1}$$ so that $$\int_0^t2ke^{-2x}\left(1-e^{-2x}\right)^{k-1}dx=\left(1-e^{-2x}\right)^k |_0^t=\left(1-e^{-2t}\right)^k.$$