Integral of $\int\frac{1}{4x^2+4x+2}dx$

Question

Solve: $$\int\frac{1}{4x^2+4x+2}dx$$

I think it may related to $\tan^{-1}x$. Any hint?

Answer 1

Hint: The denominator turns into $$(2x+1)^2+1$$

so you can make the substitution $$y = 2x+1$$ to turn the integral into

$$\frac{1}{2}\int \frac{1}{y^2+1}\ dy$$

Then try to make a trigonometric substitution.