Λ = ∫ x^3 (x^3 +1)^(10) dx


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Question Number 136067 by bramlexs22 last updated on 18/Mar/21

$Λ = \int x^{3} {(x^{3} + 1)}^{10} d x$
	Answered by Olaf last updated on 18/Mar/21

	$Λ = \int x^{3} {(x^{3} + 1)}^{10} d x$ $Λ = \int x^{3} \underset{k = 0}{\sum^{10}} C_{k}^{10} x^{3 k} d x$ $Λ = \int \underset{k = 0}{\sum^{10}} C_{k}^{10} x^{3 k + 3} d x$ $Λ = \underset{k = 0}{\sum^{10}} \frac{C_{k}^{10}}{3 k + 4} x^{3 k + 4} + C$ $Λ = \frac{1}{34} x^{34} + \frac{10}{31} x^{31} + \frac{45}{28} x^{28} + \frac{24}{5} x^{25} + \frac{105}{11} x^{22}$ $+ \frac{252}{19} x^{19} + \frac{105}{8} x^{16} + \frac{120}{13} x^{13} + \frac{9}{2} x^{10} + \frac{10}{7} x^{7}$ $+ \frac{1}{4} x^{4} + C$
	Commented by bramlexs22 last updated on 19/Mar/21

	$y e s . . . . t h a n k s$
	Answered by Ñï= last updated on 18/Mar/21
	$Λ=∫x^3 (x^3 +1)^(10) dx =(1/(33))∫xd[(x^3 +1)^(11) ] =(1/(33)){(x^3 +1)^(11) x−∫(x^3 +1)^(11) dx} =(1/(33))(x^3 +1)^(11) x−(1/(33))Σ_(n=0) ^(11) (((11)),(n) )∫x^3 dx =(1/(33))(x^3 +1)^(11) x−(1/(132))Σ_(n=0) ^(11) (((11)),(n) )x^4 +C$
	$Λ = \int x^{3} {(x^{3} + 1)}^{10} d x$ $= \frac{1}{33} \int x d [{(x^{3} + 1)}^{11}]$ $= \frac{1}{33} {{(x^{3} + 1)}^{11} x - \int {(x^{3} + 1)}^{11} d x}$ $= \frac{1}{33} {(x^{3} + 1)}^{11} x - \frac{1}{33} \underset{n = 0}{\sum^{11}} (\begin{matrix} 11 \\ n \end{matrix}) \int x^{3} d x$ $= \frac{1}{33} {(x^{3} + 1)}^{11} x - \frac{1}{132} \underset{n = 0}{\sum^{11}} (\begin{matrix} 11 \\ n \end{matrix}) x^{4} + C$
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