Integrate $\int \frac{2x}{x^2 +2x+1}$ Here is my attempt:
$$ \frac{2x}{x^2 +2x+1}= \frac{2x}{(x+1)^2 } = \frac{2}{x+1}-\frac{2}{(x+1)^2 }$$
Then I tried to integrate it,I got $2\ln(x+1)+\frac{2}{x+1}+C$ as my answer. Am I right? please correct me if I'm wrong.

Your answer is almost correct, but you should put modulus sign in your log's argument as it must be positive i.e your answer should be;
$$2\ln|x+1|+\frac{2}{x+1}+C$$
You can try one more method too.
$$\int \frac{2x}{x^2+2x+1}dx$$
Try to create derivative of denominator in numerator
$$\int \frac{2x+2-2}{x^2+2x+1}dx$$
$$\int \frac{2x+2}{x^2+2x+1}dx-2\int \frac{1}{x^2+2x+1}dx$$
$$\int \frac{2x+2}{x^2+2x+1}dx-2\int \frac{1}{(x+1)^2}dx$$
For first integral simply take $x^2+2x+1=t$, then $(2x+2)dx=dt$, hence you will get:
$$\ln(x^2+2x+1)+\frac{2}{x+1}+C$$
$$\ln(x+1)^2+\frac{2}{x+1}+C$$
$$2\ln|x+1|+\frac{2}{x+1}+C$$
