Question

Integral dan turunan dari akar (1 - akar(x))

Answer 1

Integral :

∫√(1 - √(x)) dx

misal,

u = 1 - √x
u - 1 = -√x
(u - 1)² = (-√x)²
(u - 1)² = x
2(u - 1) du = dx

∫√(1 - √x)) dx = ∫√u.2(u - 1) du
= ∫√u.(2u - 2) du
= ∫2u^(3/2) - 2u^(1/2) du
= (4/5)u^(5/2) - (4/3)u^(3/2) + C
= (4/5)u²√u - (4/3)u.√u + C
= (4/5)(1 - √x)²√(1 - √x) - (4/3)(1 - √x)√(1 - √x) + C

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Turunan, gunakan aturan rantai.

√(1 - √(x))

misal, y = √(1 - √x)

a = 1 - √x
da/dx = - 1/(2√x)

y = √a
dy/da = 1/(2√a)

dy/dx = dy/da * da/dx
= 1/(2√a) * (-1/(2√x))
= -1/(2√a.2√x)
= -1/(4.√(1 - √x).√x)
= -1/(4√(x - x√x))