Chapter 5 - Inner Product Spaces - 5.3 Orthonormal Bases: Gram-Schmidt Process - 5.3 Exercises - Page 257: 50

see the details below.

We have $$\langle x^2-\frac{1}{3}, x^2-\frac{1}{3} \rangle=\int_{-1}^1 (x^2-\frac{1}{3})^2 d x=\int_{-1}^1 x^4-\frac{2}{3}x^2+ \frac{1}{9} d x\\=\left[ \frac{ 1}{5 } x^5-\frac{2}{9}x^3+\frac{1}{9}x\right]_{-1}^1=\frac{ 2}{5 }-\frac{ 4}{9 }+\frac{2}{9} =\frac{ 8}{45 }.$$