Understanding when to use different methods of integration

Question

I'm trying to do some textbook problems, but I think there is one pivotal part I'm not understanding, because putting these methods of integration is not working for me.

For example:

given the problem

$$\int \frac{dx}{\sqrt {x^2 + 2x + 5}}$$

I complete the square, giving me

$$\int \frac{dx}{\sqrt {(x + 1)^2 + 4}}$$

This is where I lose understanding. I would simply reduce this to

$$\int \frac{dx}{(x + 1) + 2}$$

But an online calculator (Symbolab) would dictate I follow U-substitution, then substitute tan in for u, and so forth. I know there is something I don't understand, but I'm not sure what. Any help is appreciated, thank you!

Answer 1

If you want to understand calculus, you need to learn the basics. It should be totally clear to you that $\sqrt{a+b} \neq \sqrt a + \sqrt b$, as already mentioned in the commends.

Hint: Completing the square is a good idea. If you think of $$ \int \frac{1}{\sqrt{1-x^2}} \; \mathrm d x= \arcsin x + C$$ how can you manipulate (substitute) to get there? You might want to factor out $2=\sqrt 4$ in the integral $$\int \frac{1}{\sqrt {(x + 1)^2 + 4}} \:\mathrm d x$$ first.

Answer 2

set $$\sqrt{x^2+2x+5}=t+x$$ to compute the integral squaring the whole equation and solving for $x$ we get $$x=\frac{t^2-5}{2-2t}$$ then you can compute $$dx$$ $$dx=-1/2\,{\frac {{t}^{2}-2\,t+5}{ \left( -1+t \right) ^{2}}}dt$$