Find where they intersect
16 / (1 + x^4) = 8x^2
16 = 8 * x^2 * (1 + x^4)
2 = x^2 * (1 + x^4)
x^2 = t
2 = t * (1 + t^2)
2 = t + t^3
t = 1 is a solution, so t - 1 is a root
t^3 + t - 2 = 0
(t - 1) * (at^2 + bt + c) = 0
(t - 1) * (at^2 + bt + c) = t^3 + t - 2
at^3 + bt^2 + ct - at^2 - bt - c = t^3 + 0t^2 + t - 2
a = 1
b - a = 0
c - b = 1
-c = -2
b = a
b = 1
c - 1 = 1
c = 2
(t - 1) * (t^2 + t + 2) = 0
t^2 + t + 2 = 0
t = (-1 +/- sqrt(1 - 8)) / 2
t = (-1 +/- i * sqrt(7)) / 2
t = x^2
t = 1
1 = x^2
-1 , 1 = x
Confirm that both are continuous and differentiable along the interval. They are. Find which one is greater
f(0) = 8 * 0^2 = 0
g(0) = 16 / (1 + 0^4) = 16
16 / (1 + x^4) > 8x^2 along the interval
Now we integrate
16 * dx / (1 + x^4) - 8 * x^2 * dx
You can follow the steps of integrating 16 * dx / (1 + x^4) here:
https://www.symbolab.com/solver/indefinite-integral-calculator/%5Cint%20%5Cleft(%5Cfrac%7B16%7D%7B1%20%2B%20x%5E%7B4%7D%7D%5Cright)dx
Let's just apply our limits
2 * sqrt(2) * (ln|x^2 + sqrt(2) * x + 1| - ln|x^2 - sqrt(2) * x + 1| - 2 * arctan(1 - sqrt(2) * x) + 2 * arctan(1 + sqrt(2) * x))
2 * sqrt(2) * (ln|1 + sqrt(2) + 1| - ln|1 - sqrt(2) + 1| - 2 * arctan(1 - sqrt(2)) + 2 * arctan(1 + sqrt(2))) - 2 * sqrt(2) * (ln|1 - sqrt(2) + 1| - ln|1 + sqrt(2) + 1| - 2 * arctan(1 + sqrt(2)) + 2 * arctan(1 - sqrt(2))) =>
2 * sqrt(2) * (ln|2 + sqrt(2)| - ln|2 - sqrt(2)| - ln|2 - sqrt(2)| + ln|2 + sqrt(2)| + 2 * arctan(1 + sqrt(2)) - 2 * arctan(1 - sqrt(2)) + 2 * arctan(1 + sqrt(2)) - 2 * arctan(1 - sqrt(2)) =>
2 * sqrt(2) * (2 * ln|2 + sqrt(2)| - 2 * ln|2 - sqrt(2)| + 4 * arctan(1 + sqrt(2)) - 4 * arctan(1 - sqrt(2))) =>
2 * sqrt(2) * 2 * (ln|2 + sqrt(2)| - ln|2 - sqrt(2)| + 2 * (arctan(1 + sqrt(2)) - arctan(1 - sqrt(2)))) =>
4 * sqrt(2) * (ln|(2 + sqrt(2)) / (2 - sqrt(2))| + 2 * (arctan(1 + sqrt(2)) - arctan(1 - sqrt(2)))) =>
4 * sqrt(2) * (ln|(4 + 4 * sqrt(2) + 2) / (4 - 2)| + 2 * (arctan(1 + sqrt(2)) - arctan(1 - sqrt(2)))) =>
4 * sqrt(2) * (ln|3 + 2 * sqrt(2)| + 2 * (arctan(1 + sqrt(2)) - arctan(1 - sqrt(2))))
k = arctan(1 + sqrt(2)) - arctan(1 - sqrt(2))
k = arctan(a) - arctan(b)
tan(k) = tan(arctan(a) - arctan(b))
tan(k) = (tan(arctan(a)) - tan(arctan(b)) / (1 + tan(arctan(a)) * tan(arctan(b)))
tan(k) = (a - b) / (1 + a * b)
tan(k) = (1 + sqrt(2) - 1 + sqrt(2)) / (1 + (1 - 2)))
tan(k) = 2 * sqrt(2) / 0
tan(k) is undefined, so k = pi/2 (or 3pi/2)
4 * sqrt(2) * (ln|3 + 2 * sqrt(2)| + 2 * (arctan(1 + sqrt(2)) - arctan(1 - sqrt(2)))) =>
4 * sqrt(2) * (ln|3 + 2 * sqrt(2)| + 2 * (pi/2)) =>
4 * sqrt(2) * (ln|3 + 2 * sqrt(2)| + pi)