Untuk x =/= 0, dan y=akar(x^2 akar(3 akar(x^2 akar(3)))),

(4/9) * 3^(5/16) * x^(9/4) + C

Pembahasan

Kita diminta untuk menentukan hasil dari integral y dx, di mana y = akar(x^2 akar(3 akar(x^2 akar(3)))) dan x ≠ 0.

Mari kita sederhanakan ekspresi untuk y:
y = (x^2 * (3 * (x^2 * (3)^(1/2))^(1/2))^(1/2))^(1/2)

Kita bisa menyederhanakan bagian dalam akar terlebih dahulu:
 akar(3) = 3^(1/2)
 x^2 akar(3) = x^2 * 3^(1/2)
 akar(x^2 akar(3)) = (x^2 * 3^(1/2))^(1/2) = x^(2/2) * 3^((1/2)*(1/2)) = x * 3^(1/4)
 3 akar(x^2 akar(3)) = 3 * (x * 3^(1/4)) = 3^(1 + 1/4) * x = 3^(5/4) * x
 akar(3 akar(x^2 akar(3))) = (3^(5/4) * x)^(1/2) = 3^((5/4)*(1/2)) * x^(1/2) = 3^(5/8) * x^(1/2)
 x^2 akar(3 akar(x^2 akar(3))) = x^2 * 3^(5/8) * x^(1/2) = 3^(5/8) * x^(2 + 1/2) = 3^(5/8) * x^(5/2)
 akar(x^2 akar(3 akar(x^2 akar(3)))) = (3^(5/8) * x^(5/2))^(1/2) = 3^((5/8)*(1/2)) * x^((5/2)*(1/2)) = 3^(5/16) * x^(5/4)

Jadi, y = 3^(5/16) * x^(5/4).

Sekarang kita akan mengintegralkan y dx:
Integral y dx = Integral (3^(5/16) * x^(5/4)) dx

Karena 3^(5/16) adalah konstanta, kita bisa mengeluarkannya dari integral:
= 3^(5/16) * Integral (x^(5/4)) dx

Menggunakan aturan pangkat untuk integral: Integral x^n dx = (x^(n+1)) / (n+1) + C
Di sini, n = 5/4.
Integral (x^(5/4)) dx = (x^(5/4 + 1)) / (5/4 + 1)
= (x^(9/4)) / (9/4)
= (4/9) * x^(9/4)

Jadi, hasil integralnya adalah:
= 3^(5/16) * (4/9) * x^(9/4) + C
= (4/9) * 3^(5/16) * x^(9/4) + C