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y=(x²-1)(x³+3) ...

given u = cos(x), therefore du/dx=-sin(x), as tan(x)=sin(x)/cos(x), can rewrite tan(x)=(-du/dx)/u, therefore integral can become [(-1/u)du], after inegrating you are left with -ln(u)+c, therefore ln(1/u)+...

Therefore, y = x^xcan then natural log both sides leaving ln(y) = xln(x) then differentiating both sides wrst to x d/dx(ln(y)=xln(x))we are then left with this expression (dy/dx)(1/y)=ln(x)+1 mu...

Resolving all forces in the vertical gives the normal force as 10gcos(30). Resolving all forces normal to the slope gives the frictional force as 2gcos(30) (given Friction = mu*R). Using second law, F=ma....

To find an extreme point of a function you must first take the derivative of f(x) with respect to x.
As the function will peak/trough at the extreme point, the gradient at this point will be equal to...