Forget how to apply integration techniques to this question

Forget how to apply integration techniques to this question. Bonus feedback for those explanation behind answers. Helps to re jog the memory

Solution

Solve the separable equation ( dy(x))/( dx) = -2 y(x)-x^2 y(x)+x y(x):
Simplify:
( dy(x))/( dx) = (-x^2+x-2) y(x)
Divide both sides by y(x):
(( dy(x))/( dx))/(y(x)) = -x^2+x-2
Integrate both sides with respect to x:
integral (( dy(x))/( dx))/(y(x)) dx = integral (-x^2+x-2) dx
Evaluate the integrals:
log(y(x)) = -x^3/3+x^2/2-2 x+c_1, where c_1 is an arbitrary constant.
Solve for y(x):
y(x) = e^(-x^3/3+x^2/2-2 x+c_1)
Simplify the arbitrary constant:
y(x) = c_1 e^(-x^3/3+x^2/2-2 x)

2.)Solve the separable equation ( dy(x))/( dx) = x (y(x)^2+1):
Divide both sides by y(x)^2+1:
(( dy(x))/( dx))/(y(x)^2+1) = x
Integrate both sides with respect to x:
integral (( dy(x))/( dx))/(y(x)^2+1) dx = integral x dx
Evaluate the integrals:
tan^(-1)(y(x)) = x^2/2+c_1, where c_1 is an arbitrary constant.
Solve for y(x):
y(x) = tan(x^2/2+c_1);
puty(1)=0 c_1=-0.5;
y(x)= tan(x^2/2-0.5)