Prove that \( y=\frac{1}{x^{2}-1} \) is a solution for this differential \[ 2 x y^{2}+y^{\prime}=0 \]

What is the general solution to the following: \( \int 2 x \sin (x) d x \) ?

20. \( \int x \tan ^{2} x d x \)

4. Let \( f(x)=\ln \left(x^{2}+1\right) \), then \( f^{\prime}(x)= \) Let \( f(x)=e^{x} \), then \( f^{n}(x)= \) The function \( f(x)=-2 x^{3}+a x^{2}+b x+c \) the turnig point of \( p(8,-9) \) and \( (5,18) \), then the value of \( a= \) and \( b= \).

Integrate the following: \( \int x e^{2 x} d x \)

What is the solution to the differential equation \( \frac{d y}{d x}+x y=x, y(0)=-6 ? \)