Q.Using second fundamental theorem, evaluate $\int_{-1}^{1} \frac{2x + 3}{x^2 + 3x + 7}\, dx$.

Since $\frac{d}{dx}(x^2+3x+7)=2x+3$, the integral is $\ln|x^2+3x+7|$ evaluated from $-1$ to $1$, giving $\ln(11/5)$.

Here $\dfrac{d}{dx}\left(x^2+3x+7\right)=2x+3$, which is the numerator. Using $\int\dfrac{f'(x)}{f(x)}dx=\ln|f(x)|$,

$\int_{-1}^{1}\frac{2x+3}{x^2+3x+7}\,dx=\Big[\ln\big(x^2+3x+7\big)\Big]_{-1}^{1}.$ …