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Let \( f(\alpha) \) denote the area of the region in the first quadrant bounded by \( x=0, x=1, y^{2}=x \) and \( y=|\alpha x-5|-|1-\alpha x|+\alpha x^{2} \). Then \( \left(f(0)+f(1)\right) \) is equal to:

If $\int \left( \frac{1 - 5\cos^2 x}{\sin^5 x \cos^2 x} \right) dx = f(x) + C$, where $C$ is the constant of integration, then $f\left(\frac{\pi}{6}\right) - f\left(\frac{\pi}{4}\right)$ is equal to:

Let \( f(x) = \int \frac{(2 - x^2)e^x}{\sqrt{1 + x}(1 - x)^{3/2}} \, dx \). If \( f(0) = 0 \), then \( f\!\left(\frac{1}{2}\right) \) is equal to:

If $\int (\sin x)^{-\frac{11}{2}} (\cos x)^{-\frac{5}{2}} dx = -\frac{p_1}{q_1}(\cot x)^{\frac{9}{2}} - \frac{p_2}{q_2}(\cot x)^{\frac{5}{2}} - \frac{p_3}{q_3}(\cot x)^{\frac{1}{2}} + \frac{p_4}{q_4}(\cot x)^{-\frac{3}{2}} + C$, where $p_i$ and $q_i$ are positive integers with $\gcd(p_i, q_i)=1$ for $i=1,2,3,4$ and $C$ is the constant of integration, then $\frac{15p_1p_2p_3p_4}{q_1q_2q_3q_4}$ is equal to _________.

\( f(x)=\displaystyle \int \dfrac{7x^{10}+9x^{8}}{(1+x^{2}+2x^{9})^{2}} \, dx,\ x>0, \ \lim_{x\to 0} f(x)=0 \) and \( f(1)=\dfrac{1}{4} \). . and \( B=\operatorname{adj}(\operatorname{adj}A) \) be such that \( |B|=81 \), then \( \alpha^{2} \) is equal to:

Let \( f(t)=\int \frac{(1-\sin (\log_{e} t))}{(1-\cos (\log_{e} t))}\,dt,\; t>1 \). If \( f\!\left(e^{\pi/2}\right)=-e^{-\pi/2} \) and \( f\!\left(e^{\pi/4}\right)=\alpha e^{\pi/4} \), then \( \alpha \) equals:

Let \( f:\mathbb{R}\to \mathbb{R} \) be a differentiable function such that \( f\!\left(\frac{x+y}{3}\right)=\frac{f(x)+f(y)}{3} \) for all \( x,y\in \mathbb{R} \), and \( f'(0)=3 \). Then the minimum value of the function \( g(x)=3+e^{x}f(x) \), is:

Let \( f(x)=\displaystyle \int \left(\dfrac{16x+24}{x^{2}+2x-15}\right)dx \). 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: