\( \frac { y } { d x } + \frac { y \cos x + \sin y + y } { \sin x + x \cos y + x } \)

Calculate the integral of \( f(x, y)=\left(x^{2}+y^{2}\right)^{-2} \) over the region \( x^{2}+y^{2} \leq 25, x \geq \frac{5}{\sqrt{2}} \) by changing to polar coordinates. (Use symbolic notation and fractions where needed.) \( \iint_{D} f(r, \theta) d r d \theta=\square \) help (fractions).

25. \( N=\frac{p+q}{p+r}, \quad p=u+v w, \quad q=v+u w, \quad r=w+u v \) \( \frac{\partial N}{\partial u}, \frac{\partial N}{\partial v}, \frac{\partial N}{\partial w} \quad \) donde \( u=2, v=3, w=4 \)

3. Sea \( f(x)=\cos (x) \), es continua en el intervalo \( [0,2 \pi] \), si \( x \) un punto dentro del intervalo. Encuentre \( F(x)= \)

Find the minimum value of \( f(t)=2 t^{3}-24 t^{2}+90, t \geq 0 \), and give the value of \( t \) where this minimum occurs. The minimum value occurs when \( t=\square \) with a value of \( \square \).

25. \( N=\frac{p+q}{p+r}, \quad p=u+v w, \quad q=v+u w, \quad r=w+u v \) \( \frac{\partial N}{\partial u}, \frac{\partial N}{\partial v}, \frac{\partial N}{\partial w} \) donde \( u=2, v=3, w=4 \)