I need to solve the definite integral: $$\int_{0}^{2} \sqrt{(1+x)\sqrt{4x+1}-3x+1}dx$$ The integral was proposed by my algebraic geometry professor as a warm up excercise, he hinted us to research about elliptic functions and curves, but I cannot find anything related to this integral. I tried substition and integration by parts, by I can't seem to reduce the problem.
I already solved the integral already by proving the convergence of it Taylor Series around $x_0=2$ and integrating said power series. But, its only an approximation, since I can't integrate infinite terms.
Is it possible to solve it analytically? What am I missing?
Best Answer
For a nightmare, look at the antiderivative given by Wolfram Alpha. If you like elliptic integrals and complex numbers, I suppose that you are more than happy.
Using the integration bounds, the result is just a monster. Numerically, $$I=3.23567625418595545329101113302458555831569511979865687\cdots$$
Interesting (for the fun) is that an inverse symbolic calculator proposes, as an approximation, the reciprocal of the smallest root of the cubic equation $$15572 x^3+13982 x^2-78333 x+22414=0$$
This is in a relative error of $3.1\times 10^{-19}$.