Where was my mistake (integration by trig-substitution problem)?

Question

I am attempting to solve the problem

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

First, I complete the square, then factor out a $\frac {3}{4}$:

$$\int \frac{dx}{\frac{3}{4}(\frac{4}{3}(x+\frac{1}{2})^2+1)}$$

Let $u = \sqrt{\frac{4}{3}}(x+\frac{1}{2})$

$$\frac{du}{dx} = \frac{2}{\sqrt{3}}$$

$$dx = \frac{\sqrt{3}}{2} du$$

Thus, we now have the integral:

$$\frac{4}{3} \frac{\sqrt{3}}{2} \int \frac{du}{u^2+1}$$

Let $u = \tan \theta$

$$du = \sec^2\theta \ d\theta$$

What follows is obvious now, and the solution should be:

$$\frac{4}{3} \frac{\sqrt{3}}{2} \theta + C$$

$$\theta = \tan^{-1}(u)$$

Thus, the final solution is:

$$\frac{4}{3} \frac{\sqrt{3}}{2} \tan^{-1} \left( \sqrt{\frac 4 3} \left(x+\frac 1 2 \right)\right) + C$$

However, according to online calculator integral-calculator, the answer is:

$$\frac 2 {\sqrt 3} \tan^{-1} \left( \frac{2x+1}{\sqrt 3} \right)+C$$

Any indication as to where my mistake falls would be very beneficial.

Answer 1

This answer was posted prior to an edit made by the OP.

Note that we have

$$x^2+x+1=(x+1/2)^2+3/4\ne \frac34 \left(\frac43 (x+1/2)^2+\frac34 \right)=x^2+2+5/2$$

Answer 2

Your only mistake appears to be a failure to notice that $\displaystyle \frac{4}{3} \frac{\sqrt{3}}{2} \tan^{-1} \left( \sqrt{\frac 4 3} \left(x+\frac 1 2 \right)\right) $ is exactly the same thing as $ \displaystyle \frac 2 {\sqrt 3} \tan^{-1} \left( \frac{2x+1}{\sqrt 3} \right).$

First you have $$ \frac 4 3 \cdot \frac{\sqrt 3} 2 = \frac{4\sqrt 3}{\sqrt 3\sqrt 3 \cdot 2} = \frac 2 {\sqrt 3}. $$ And then $$ \sqrt{\frac 4 3} \left( x + \frac 1 2 \right) = \frac 2 {\sqrt 3} \left( x + \frac 1 2 \right) = \frac 1 {\sqrt 3} (2x+1). $$