The value of $\int\limits_{0}^{20\pi} (\sin^4 x + \cos^4 x) dx$ is equal to:

Let $f(x)$ be a polynomial of degree 5, and have extrema at $x = 1$ and $x = -1$. If $\lim\limits_{x \to 0} \left( \frac{f(x)}{x^3} \right) = -5$, then $f(2) - f(-2)$ is equal to:

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

Let $x = x(y)$ be the solution of the differential equation $2y^2 \frac{dx}{dy} - 2xy + x^2 = 0$, $y > 1$, $x(e) = e$.

Then $x(e^2)$ is equal to: