TITLE: How to evaluate $ \int_{0}^2 x^{26} (x-1)^{17} (5x-3)dx ~~ ?$
QUESTION [5 upvotes]: How to evaluate $$ \int_{0}^2 x^{26} (x-1)^{17} (5x-3)dx ~~ ?$$
I have tried to evaluate, $$ I = \int_{0}^2 x^{26} (x-1)^{17} (5x-3)dx = 5\int_{0}^2 x^{27} (x-1)^{17} dx - 3\int_{0}^2 x^{26} (x-1)^{17} dx$$ by parts but I am getting very lengthy expressions for both the integrals . Please help.

REPLY [2 votes]: There is a trick, the numbers have been "arranged":
$$(x^{27}(x-1)^{18})'=27x^{26}(x-1)^{17}+18x^{27}(x-1)^{16}=(27(x-1)+18x)x^{26}(x-1)^{17}\\=9(5x-3)x^{26}(x-1)^{17}.$$