I was recently asked by Mike Grigsby to integrate:
∫ √x √(1-x) dx
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Make the substitution:
x = sin^2 θ
Note:
√x = sin θ ,
θ = asin x ,
dx = 2 sin θ cos θ dθ ,
and
cos θ = √(1 - sin^2 θ )
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Hence:
∫ √x √(1-x) dx (x to θ)
= ∫ sin θ * √(1 - sin^2 θ )* 2 *sin θ * cos θ dθ
= 2 ∫ sin^2 θ cos^2 θ dθ
= 2 ∫ (1 - cos^2 θ ) * cos^2 θ dθ
= 2 ∫ cos^2 θ - cos^4 θ dθ
= 2 ∫ 1/8 - 1/8 * cos (4θ) dθ (See Note A)
= 2 ( θ/8 - 1/8 * 1/4* sin (4θ))
= 2 ( θ/8 - 1/32 * sin (4θ))
= 2 ( θ / 8 - 1/32 * [8 sin θ cos^3 θ - 4 sin θ cos θ]) (See Note B)
(θ to x)
= 2 ( asin √x / 8 - 1/32 * [8 √x (1 - x)^3/2 - 4 √x √(1-x)] )
= 2 ( asin √x / 8 - (√x √(1-x) (2 - 2x - 1))/8 )
= asin √x / 4 - (√x √(1-x) (1 - 2x)) / 4
Final:
∫ √x √(1-x) dx = 1/4 * ( asin √x - (√x √(1-x) (1 - 2x)) + C
for some constant C
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Note A
cos 2a = 2 cos^2 a - 1
cos^2 a = 1/2 * (cos 2a + 1)
cos^4 a
= [1/2 * (cos 2a + 1)]^2
= 1/4 * (cos^2 2a + 2 cos 2a + 1)
= 1/4 * ( (cos 4a + 1)/2 + 2 cos 2a + 1)
= 1/8 * (cos 4a + 4 cos 2a + 3)
= 1/8 * cos 4a + 1/2 * cos 2a + 3/8
cos^2 a - cos^4 a
= (1/2 * cos 2a + 1/2) - (1/8 * cos 4a + 1/2 * cos 2a + 3/8)
= 1/8 - 1/8 * cos 4a
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Note B
sin 2a = 2 sin a cos a
sin 4a
= 2 sin 2a cos 2a
= 8 sin a cos^3 a - 4 sin a cos a
Using the triangle from above:
sin θ = √x
cos θ = √(1-x)
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This blog is property of Edward Shore. © 2012