Solving Physical Pendulum Homework Equation

[shyta]

Homework Statement

http://hyperphysics.phy-astr.gsu.edu/hbase/pendp.html#c2

i'm trying to prove

to be

Homework Equations

Letting d = L_cm
now we already know $$\partial^2\vartheta$$/$$\partial t^2$$ = $$\alpha$$ = mgdT$$\vartheta$$ / I


I tried integrating the whole equation wrt dt

so $$\partial\vartheta$$/$$\partial t$$ = $$\int mgd\vartheta/ I dt$$ (with limits 0->T) = mgd$$\vartheta$$/ I

I only need help with this step. How do I deal with the $$\vartheta$$?

 

[zorro]

I have a better method-

Let l be the distance of COM from the axis of rotation. For the equilibrium, the COM should be vertically below the axis.
Now let us rotate the body through an angle θ. The torque of forces acting on the body about the axis is τ= -mglsinθ (The only force is due to its weight).
τ = Iα and for small angles sinθ is app. θ
so α = -mglθ/I
Comparing it with the equation of angular SHM,
α = -ω²θ,
T = 2π/ω = 2π(I/mgl)^0.5