Knowing that $\displaystyle f'(x)=\frac{x+2}{x^2+4x+11}$ and $\displaystyle f(1)=0$ then the value of $\displaystyle f(0)$ is ?
1. Calculate F(x):
$\displaystyle F(x)=\dfrac12 \cdot \int \left(\dfrac{2x+4}{x^2+4x+11}\right) dx=\dfrac12 \ln(x^2+4x+11)+C$
2. Calculate F(1):
$\displaystyle F(1)=\dfrac12 \ln(1+4+11)+C=2\ln(2)+C = 0~\implies~ C=-2\ln(2)$
3. Calculate F(0):
I'll leave this part for you.