Question

[
begin{array}{l}
int_{-1}^{1} f_{n}(x) P_{n}(x) d x=2(-1)^{n} frac{a_{n}}{2^{n}}
int_{0}^{1}left(x^{2}-1right)^{n} d x=2(-1)^{n} frac{a_{n}}{2^{n}} cdot I_{n}
end{array}
]
( ldots ldots . .(6) )
ldon't understand as in shouldnt it be like this,
[
begin{array}{c}
int_{-1}^{1} f_{n}(x) P_{n}(x) d x=(-1)^{n} frac{a_{n}}{2^{n}}
int_{-1}^{1}left(x^{2}-1right)^{n} d x=0
end{array}
]
as they should cancel out even if the integral is non-zero.

# 9. (a) State and prove orthogonal properties of Legendre polynomials.

Solution