∫(1/x³ - a³)dx = ?
Notice that it is the difference of two cubes and factor it using:
a³ - b³ = ( a - b )·( a² + a·b + b² )
————
∫1/(x³ - a³) dx
= ∫1/[ ( x - a )·( x² + a·x + a² ) ] dx
Use partial fractions:
1/[ ( x - a )·( x² + a·x + a² ) ] = A/( x - a ) + B/( x² + a·x + a² )
1 = A·( x² + a·x + a² ) + B·( x - a )
Choose a convenient value for x to cancel either A or B. Choose x=a:
1 = A·( (a)² + a·(a) + a² ) + B·( (a) - a )
1 = A·( 3·a² )
1/( 3·a² ) = A
Use this value to solve for B:
1 = 1/( 3·a² ) · ( x² + a·x + a² ) + B·( x - a )
-B·( x - a ) = 1/( 3·a² ) · ( x² + a·x + a² ) - 1
-B·3·a²·( x - a ) = x² + a·x - 2·a²
-B·3·a²·( x - a ) = ( x - a )·( x + 2·a )
B = -( x + 2·a ) / ( 3·a² )
∫ [ A/( x - a ) + B/( x² + a·x + a² ) ] dx
→ ∫ [ 1/( 3·a² ) /( x - a ) - ( x + 2·a ) / ( 3·a² ) / ( x² + a·x + a² ) ] dx
= 1/( 3·a² )·∫1/( x - a )dx - 1/( 3·a² )·∫( x + 2·a )/( x² + a·x + a² )dx
This are two integrals now.
(I'll leave out the factored-out constants while evaluating.)
Integral one:
∫1/( x - a ) dx
Let u = x - a
Then du = dx
→ ∫1/u du
= ln|u| + C,
Integral two:
∫( x + 2·a )/( x² + a·x + a² )]dx
= ½·∫( 2·x + a + 3·a )/( x² + a·x + a² )]dx
= ½·∫( 2·x + a )/( x² + a·x + a² )]dx + ½·∫( 3·a )/( x² + a·x + a² )]dx
Let v = x² + a·x + a²
Then dv = (2·x + a)dx
→ ½·∫dv/v + ½·∫( 3·a )/( x² + a·x + a² )]dx
= ½·ln| v | + 3·a/2·∫1/( x² + a·x + a² )]dx
Last part is sadly the hardest.
You want to make a trigonometric substitution.
Complete the square in the denominator:
∫1/( x² + a·x + a² ) dx
= ∫1/( (x² + a·x + (a/2)² - (a/2)² + a² ) dx
= ∫1/[(x + a/2)² + 3·a²/4 ] dx
Let q = x + a/2
Then dq = dx
→ ∫1/( q² + 3·a²/4 ) dq
Get 3·a²/4 to be just 1. Multiply by 1/1:
= [4/(3·a²)]/[ 4/(3·a²) ] · ∫1/( q² + 3·a²/4 ) dq
= [ 4/(3·a²) ] · ∫1/{ [4/(3·a²)]·q² + 1 ] dq
Bring the the coefficient of q² into the square by getting its square root:
= [ 4/(3·a²) ] · ∫1/{ [2/(√(3)·a)·q]² + 1 ] dq
Let tan(r) = 2/[√(3)·a]·q
Then r = arctan[ 2/[√(3)·a]·q ]
And dr = 1 / { [ 2/[√(3)·a]·q ]² + 1 } · 2/[√(3)·a] · dq
And dq = { [ 2/[√(3)·a]·q ]² + 1 } · √(3)·a/2 · dr
Replace dq and reduce:
→ [ 4/(3·a²) ] · ∫1/{ [2/(√(3))·a·q]² + 1 ]·{ [ 2/[√(3)·a]·q ]² + 1 } · √(3)·a/2·dr
= [ 4/(3·a²) ] · ∫ √(3)·a/2 dr
= 2/[ √(3)·a ] · r + C,,
Put these integrals all back together with the left-out constants:
1/( 3·a² )·∫1/( x - a )dx - 1/( 3·a² )·∫( x + 2·a )/( x² + a·x + a² )dx
= 1/( 3·a² )·{ ln|u| } - 1/( 3·a² )·{ ½·ln| v | + 3·a/2·2/[ √(3)·a ]·r } + C
Reverse the substitutions for u, v, and r (and then for q inside r):
= 1/( 3·a² )·{ ln|x - a| } - 1/( 3·a² )·{ ½·ln|x² + a·x + a²| + 3·a/2·2/[ √(3)·a ]·arctan[ 2/[√(3)·a]·q ] } + C
= 1/( 3·a² )·{ ln|x - a| } - 1/( 3·a² )·{ ½·ln|x² + a·x + a²| + 3·a/2·2/[ √(3)·a ]·arctan[ 2/[√(3)·a]·(x + a/2) ] } + C
Simplify:
= 1/(3·a²)·{ ln|x - a| - ½·ln|x² + a·x + a²| - √(3)·arctan[(2·x/a + 1)/√(3)] } + C
I am not sure,
but, let me give a try!
â«(1/x³ - a³)dx = (1/(3x^2)) x log(x^3 - a^3)
(or)
â«(1/x³ - a³)dx = log(x^3 - a^3) / (3x^2))
Both answers here are same Only!
Hello
write (x³-a³)=(x-a)(x²+ax+a²)
Then find A,B and C such that
1/(x³-a³) = A/(x-a) + (Bx+C)/(x²+ax+a²)
Integration is immediate as you can see after Gravitate's work;-)
no thanks, it is totally horrible
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Answers & Comments
Verified answer
Notice that it is the difference of two cubes and factor it using:
a³ - b³ = ( a - b )·( a² + a·b + b² )
————
∫1/(x³ - a³) dx
= ∫1/[ ( x - a )·( x² + a·x + a² ) ] dx
————
Use partial fractions:
1/[ ( x - a )·( x² + a·x + a² ) ] = A/( x - a ) + B/( x² + a·x + a² )
1 = A·( x² + a·x + a² ) + B·( x - a )
Choose a convenient value for x to cancel either A or B. Choose x=a:
1 = A·( (a)² + a·(a) + a² ) + B·( (a) - a )
1 = A·( 3·a² )
1/( 3·a² ) = A
Use this value to solve for B:
1 = 1/( 3·a² ) · ( x² + a·x + a² ) + B·( x - a )
-B·( x - a ) = 1/( 3·a² ) · ( x² + a·x + a² ) - 1
-B·3·a²·( x - a ) = x² + a·x - 2·a²
-B·3·a²·( x - a ) = ( x - a )·( x + 2·a )
B = -( x + 2·a ) / ( 3·a² )
————
∫ [ A/( x - a ) + B/( x² + a·x + a² ) ] dx
→ ∫ [ 1/( 3·a² ) /( x - a ) - ( x + 2·a ) / ( 3·a² ) / ( x² + a·x + a² ) ] dx
= 1/( 3·a² )·∫1/( x - a )dx - 1/( 3·a² )·∫( x + 2·a )/( x² + a·x + a² )dx
This are two integrals now.
(I'll leave out the factored-out constants while evaluating.)
————
Integral one:
∫1/( x - a ) dx
Let u = x - a
Then du = dx
→ ∫1/u du
= ln|u| + C,
————
Integral two:
∫( x + 2·a )/( x² + a·x + a² )]dx
= ½·∫( 2·x + a + 3·a )/( x² + a·x + a² )]dx
= ½·∫( 2·x + a )/( x² + a·x + a² )]dx + ½·∫( 3·a )/( x² + a·x + a² )]dx
Let v = x² + a·x + a²
Then dv = (2·x + a)dx
→ ½·∫dv/v + ½·∫( 3·a )/( x² + a·x + a² )]dx
= ½·ln| v | + 3·a/2·∫1/( x² + a·x + a² )]dx
————
Last part is sadly the hardest.
You want to make a trigonometric substitution.
Complete the square in the denominator:
∫1/( x² + a·x + a² ) dx
= ∫1/( (x² + a·x + (a/2)² - (a/2)² + a² ) dx
= ∫1/[(x + a/2)² + 3·a²/4 ] dx
Let q = x + a/2
Then dq = dx
→ ∫1/( q² + 3·a²/4 ) dq
Get 3·a²/4 to be just 1. Multiply by 1/1:
= [4/(3·a²)]/[ 4/(3·a²) ] · ∫1/( q² + 3·a²/4 ) dq
= [ 4/(3·a²) ] · ∫1/{ [4/(3·a²)]·q² + 1 ] dq
Bring the the coefficient of q² into the square by getting its square root:
= [ 4/(3·a²) ] · ∫1/{ [2/(√(3)·a)·q]² + 1 ] dq
Let tan(r) = 2/[√(3)·a]·q
Then r = arctan[ 2/[√(3)·a]·q ]
And dr = 1 / { [ 2/[√(3)·a]·q ]² + 1 } · 2/[√(3)·a] · dq
And dq = { [ 2/[√(3)·a]·q ]² + 1 } · √(3)·a/2 · dr
Replace dq and reduce:
→ [ 4/(3·a²) ] · ∫1/{ [2/(√(3))·a·q]² + 1 ]·{ [ 2/[√(3)·a]·q ]² + 1 } · √(3)·a/2·dr
= [ 4/(3·a²) ] · ∫ √(3)·a/2 dr
= 2/[ √(3)·a ] · r + C,,
————
Put these integrals all back together with the left-out constants:
1/( 3·a² )·∫1/( x - a )dx - 1/( 3·a² )·∫( x + 2·a )/( x² + a·x + a² )dx
= 1/( 3·a² )·{ ln|u| } - 1/( 3·a² )·{ ½·ln| v | + 3·a/2·2/[ √(3)·a ]·r } + C
Reverse the substitutions for u, v, and r (and then for q inside r):
= 1/( 3·a² )·{ ln|x - a| } - 1/( 3·a² )·{ ½·ln|x² + a·x + a²| + 3·a/2·2/[ √(3)·a ]·arctan[ 2/[√(3)·a]·q ] } + C
= 1/( 3·a² )·{ ln|x - a| } - 1/( 3·a² )·{ ½·ln|x² + a·x + a²| + 3·a/2·2/[ √(3)·a ]·arctan[ 2/[√(3)·a]·(x + a/2) ] } + C
Simplify:
= 1/(3·a²)·{ ln|x - a| - ½·ln|x² + a·x + a²| - √(3)·arctan[(2·x/a + 1)/√(3)] } + C
I am not sure,
but, let me give a try!
â«(1/x³ - a³)dx = (1/(3x^2)) x log(x^3 - a^3)
(or)
â«(1/x³ - a³)dx = log(x^3 - a^3) / (3x^2))
Both answers here are same Only!
Hello
write (x³-a³)=(x-a)(x²+ax+a²)
Then find A,B and C such that
1/(x³-a³) = A/(x-a) + (Bx+C)/(x²+ax+a²)
Integration is immediate as you can see after Gravitate's work;-)
no thanks, it is totally horrible