Partial fractions

Let $f(x)=\displaystyle\int\left(\frac{16x+24}{x^2+2x-15}\right)dx$. If $f(4)=14\log_e(3)$ and $f(7)=\log_e\left(2^\alpha\cdot3^\beta\right)$, $\alpha,\beta\in\mathbb N$, then $\alpha+\beta$ is equal to :

If $y(x)$ is the solution of the differential equation
$$x\,dy - (y^2 - 4y)\,dx = 0 \quad\text{for } x > 0, \qquad y(1) = 2,$$
and the slope of the curve $y = y(x)$ is never zero, then the value of $10\,y(\sqrt2)$ is _____.

$$\lim_{n\to\infty}\left(\frac{n^2}{\left(n^2+1\right)(n+1)}+\frac{n^2}{\left(n^2+4\right)(n+2)}+\frac{n^2}{\left(n^2+9\right)(n+3)}+\dots+\frac{n^2}{\left(n^2+n^2\right)(n+n)}\right)$$
is equal to

The value of $b>3$ for which $12\displaystyle\int_3^b\frac1{\left(x^2-1\right)\left(x^2-4\right)}dx=\log_e\left(\frac{49}{40}\right)$, is equal to ______.

Let $y=y(x)$ be the solution of the differential equation $x\left(1-x^2\right)\dfrac{dy}{dx}+\left(3x^2y-y-4x^3\right)=0$, $x>1$, with $y(2)=-2$. Then $y(3)$ is equal to

$\displaystyle\lim_{n\to\infty}\sum_{r=1}^n\frac r{2r^2-7rn+6n^2}$ is equal to :

Let $y=y(x)$ be the solution curve of the differential equation $\dfrac{dy}{dx}+\left(\dfrac{2x^2+11x+13}{x^3+6x^2+11x+6}\right)y=\dfrac{(x+3)}{x+1}$, x > −1, which passes through the point (0, 1). Then y(1) is equal to :

The value(s) of $\displaystyle\int_0^1\frac{x^4(1-x)^4}{1+x^2}\,dx$ is (are)

Let
$$I=\int\frac{e^x}{e^{4x}+e^{2x}+1}dx,\qquad J=\int\frac{e^{-x}}{e^{-4x}+e^{-2x}+1}dx.$$
Then, for an arbitrary constant $C$, the value of $J-I$ equals