Problems "integ"

Calculate numerically the intergral
```
∫_₀^¹ dx (^ln(x)/_√x) .
```
In the variational method in quantum mechanics one calculates the expectation value E[ψ] of a Hamiltonian H,
```
E[ψ] ≡ ⟨ψ|H|ψ⟩/⟨ψ|ψ⟩ ,
```
(in Dirac notation) with some trial function ψ. The trial function is then optimized to provide the minimum of the expectation value.
Consider the Hamiltonian operator for one-dimensional oscillator,
```
H_os = - (1/2) (d²/dx²) + (1/2) x² ,
```
and the trial function in the form of a Gaussian,
```
ψ_α(x) = exp(-αx²/2) ,
```
where α is the variational parameter. Calculate numerically
```
E(α) ≡ ⟨ψ_α|H_os|ψ_α⟩/⟨ψ_α|ψ_α⟩ .
```
and plot the result – you should see a minimum of E=0.5 at α=1 (why?).
Hints:
1. The norm-integral, ⟨ψ_α|ψ_α⟩, has the form (check it)
```
⟨ψ_α|ψ_α⟩ = ∫_{_-∞}^{^∞} exp(-αx²) dx .
```
  It is actually analytic, but you have to calculate it numerically.
2. The Hamiltonian-integral, ⟨ψ_α|H_os|ψ_α⟩, has the form (check it)
```
⟨ψ_α|H_os|ψ_α⟩ = ∫_{_-∞}^{^∞} (-α²x²/2 + α/2 + x²/2)*exp(-αx²) dx .
```
  It is actually also analytic, but you again have to calculate it numerically.
3. Since the integration limits are infinite, you migh want to use gsl_integration_qagi function.