Can You Solve This Very Nice Integral Problem ?

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[@Ibrahim-fm1gf]

Love your daily integrals ❤

[@slavinojunepri7648]

Nice substitution to turn the integrand into a polynomial, which is easy to integrate! This can also be solved using integration by parts starting with u=x^2 and dv=(1+2x)^(-1/3) dx.

[@Billts]

Ωραια άσκηση 😊

[@gelbkehlchen]

Solution:
∫x²/∛(1+2x)*dx =
-----------------
Substitution: u = 1+2x x = (u-1)/2 du = 2*dx dx = du/2
-----------------
= 1/8*∫(u-1)²/∛u*du = 1/8*∫(u²-2u+1)/∛u*du
= 1/8*[∫u²/∛u*du-∫2u/∛u*du+∫1/∛u*du]
= 1/8*[∫u^(5/3)*du-2*∫u^(2/3)*du+∫u^(-1/3)*du]
= 1/8*[3/8*u^(8/3)-2*3/5*u^(5/3)+3/2*u^(2/3)+C]
= 3/64*u^(8/3)-3/20*u^(5/3)+3/16*u^(2/3)+C/8
= 3/64*(1+2x)^(8/3)-3/20*(1+2x)^(5/3)+3/16*(1+2x)^(2/3)+C/8
Checking the result by deriving:
[3/64*(1+2x)^(8/3)-3/20*(1+2x)^(5/3)+3/16*(1+2x)^(2/3)+C/8]’
= 2*8/3*3/64*(1+2x)^(5/3)-2*5/3*3/20*(1+2x)^(2/3)+2*2/3*3/16*(1+2x)^(-1/3)
= 1/4*(1+2x)^(5/3)-1/2*(1+2x)^(2/3)+1/[4*∛(1+2x)]
= [(1+2x)²-2*(1+2x)+1]/[4*∛(1+2x)]
= [1+4x+4x²-2-4x+1]/[4*∛(1+2x)]
= 4x²/[4*∛(1+2x)]
= x²/∛(1+2x) everything okay!

[@ABHISHEKKUMAR-01024]

Aliter.
Let
∫ [ x² / { ³√( 1 + 2x) } ] dx = I
Then
I = ∫ [ x² / { ³√(1 + 2x) } ] dx
= (1/4) ∫ [ 4x² / { ³√(1 + 2x) } ] dx
= (1/4) ∫ [ (2x)² / { ³√(1 + 2x) } ] dx
= (1/4) ∫ [ (2x + 1 - 1)² / { ³√(1 + 2x) } ] dx
= (1/4) ∫ [ {(1 + 2x) - 1}² / { ³√(1 + 2x) } ] dx
= (1/4) ∫ [ { (1 + 2x)²
- 2 (1 + 2x) + 1 } / { ³√(1 + 2x) } ] dx
= (1/4) ∫ [ { (1 + 2x)²
- 2 (1 + 2x) + 1 } / (1 + 2x)¹/³ ] dx
= (1/4) ∫ { (1 + 2x)² / (1 + 2x)¹/³
- 2 (1 + 2x) / (1 + 2x)¹/³
+ 1 / (1 + 2x)¹/³ } dx
= (1/4) ∫ { (1 + 2x)⁵/³ - 2 (1 + 2x)²/³
+ (1 + 2x) - ¹/³ } dx
= (1/4) [ ∫ (1 + 2x)⁵/³ dx - ∫ 2 (1 + 2x)²/³ dx
+ ∫ (1 + 2x) - ¹/³ dx ]
= (1/4) [ ∫ (1 + 2x)⁵/³ dx - 2 ∫ (1 + 2x)²/³ dx
+ ∫ (1 + 2x) - ¹/³ dx ]
= (1/4) ∫ (1 + 2x)⁵/³ dx
- (1/2) ∫ (1 + 2x)²/³ dx
+ (1/4) ∫ (1 + 2x) - ¹/³ dx
= (¹/⁴) (¹/²) [ { (1 + 2x)⁸/³ } / (⁸/³) ]
- (¹/²) (¹/²) [ { (1 + 2x)⁵/³ } / (⁵/³) ]
+ (¹/⁴) (¹/²) [ { (1 + 2x)²/³ } / (²/³) ]
= (¹/⁴) (¹/²) (³/⁸) (1 + 2x)⁸/³
- (¹/²) (¹/²) (³/⁵) (1 + 2x)⁵/³
+ (¹/⁴) (¹/²) (³/²) (1 + 2x)²/³
= (³/⁶⁴) (1 + 2x)⁸/³
- (³/²⁰) (1 + 2x)⁵/³
+ (³/¹⁶) (1 + 2x)²/³ + C