# Proof associated Legendre polynomials are orthogonal: integral doesn’t solve

Wondering why Mathematica can’t solve this integral:

Integrate[  LegendreP[l1, 1, x] *    LegendreP[l2, 1, x], {x, -1, 1}] 

Mathematica outputs: $$\int_{-1}^1 P_{\text{l1}}^1(x) P_{\text{l2}}^1(x) \, dx$$

But I see there is an analytical solution: $$\int_{-1}^{1} P^{m}_{l}(x) P^{m}_{k}(x)dx=\frac{2}{2l + 1}\frac{(l + m)!}{(l – m)!}\delta_{lk}$$ Where $$\delta_{lk}$$ is Kronecker delta

The solution looks like it involves integration by parts and trig substitution. I’m wondering why Mathematica can’t solve it, or if there is some way to modify the input so that Mathematica can figure it out. Trying to build intuition on how to use Mathematica and what are it’s limits.

I also tried forming the function myself but it didn’t help:

p[x_, l_, m_] := ((-1)^m/(2^l l!)*(1 - x^2)^(m/2) * D[(y^2 - 1)^l, {y, l + m}]) /. y -> x; Integrate[p[x, l1, 1] * p[x, l2, 1], {x, -1, 1}] 
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