If $\int \frac{2 x^{2}+3}{\left(x^{2}-1\right)\left(x^{2}+4\right)} d x=a \log…
If $\int \frac{2 x^{2}+3}{\left(x^{2}-1\right)\left(x^{2}+4\right)} d x=a \log \left|\frac{x-1}{x+1}\right|+b \tan ^{-1}\left(\frac{x}{2}\right)+C$, then
$\begin{aligned}
\text {Let } I &=\int \frac{2 x^{2}+3}{\left(x^{2}-1\right)\left(x^{2}+4\right)} d x=\int\left[\frac{1}{x^{2}-1}+\frac{1}{x^{2}+4}\right] d x \\
I &=\frac{1}{2(1)} \log \left|\frac{x-1}{x+1}\right|+\frac{1}{2} \tan ^{-1} \frac{x}{2}+C
\end{aligned}$
Comparing with given data, we get $a=\frac{1}{2}, b=\frac{1}{2}$