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Maths
A Level
Integrate 2x/(x^2+3) using the substitution u=x^2+3
u=x2 + 3
du/dx=2x
dx=du/2x
2x/(x2+3) dx becomes (2x/u) * (du/2x)
the 2x terms cancel out giving 1/u du
this integrates to ln(u)+c becoming ln(x
A Curve has parametric equation x=2sin(t), y= 1+cos(2t), -pi/2<=t<=pi/2. a) Find dy/dx when t=pi/3. b) Find the Cartesian equation for the curve in form y=f(x), -k<=x<=k. c) Find the range of f(x)
x=2sin(t), y=1+cos(2t)
a) By chain rule, dy/dx = (dy/dt)/(dx/dt)
dy/dt = -2sin(2t), dx/dt= 2cos(t)
dy/dx= -sin(2t)/cos(t)
dy/dx=-2sin(t)cos(t)/cos(t)
dy/dx=-2sin(t)
