Trying to work through a problem that requires us to complete the square, then use a sinh substitution...but I need to start by remembering back to math from many years ago.
$$\int\sqrt{2x^2+3x+4} \ dx$$
Starting to complete the square I know the goal is to arrive at the form $(x+a)^2$ so I did the following
- $$2(x^2+\frac{3}{2}x+2)$$
- $$2(x^2+2\frac{3}{4}x+\frac{9}{16})$$ because of $(x+\frac{3}{4})^2$
And then eventually I should arrive at $$(x+\frac{3}{4})^2 + \frac{23}{16}$$
Is that leap I'm making correct? I can't figure out where the 2 went on the middle term and what becomes of the 2 I took out front. My professor explained that part of this becomes easier when you subtract a $\frac{9}{16} and a 2, but they just sort of get added to the end of the line which I can't figure out how.