If $\int \frac{x+3}{(x-1)^2(2 x-1)} d x=\frac{A}{x-1}+B \log (2 x-1)+C \log (x-1)+K$,then $A+B+C=$

$\int \frac{d x}{2 e^{2 x}+3 e^x+1}=$

If $\int \frac{2x+3}{(x-1)(x^2+1)} dx = \log_e {(x-1)^{\frac{5}{2}}(x^2+1)^a} - \frac{1}{2} \tan^{-1} x + A$ where $A$ is an arbitrary constant,then the value of $a$ is

If $\int \frac{2x^2+3}{(x^2-1)(x^2-4)} dx = \log \left[\left(\frac{x-2}{x+2}\right)^a \cdot \left(\frac{x+1}{x-1}\right)^b\right] + c$,(where $c$ is the constant of integration),then the value of $a+b$ is equal to

Let $I(x) = \int \frac{(x+1)}{x(1+x e^x)^2} dx, x > 0$. If $\lim_{x \rightarrow \infty} I(x) = 0$,then $I(1)$ is equal to

Let $f(x) = \int \frac{16x + 24}{x^2 + 2x - 15} dx$. If $f(4) = 14 \log_e(3)$ and $f(7) = \log_e(2^\alpha \cdot 3^\beta)$,where $\alpha, \beta \in N$,then $\alpha + \beta$ is equal to: