x-3-x-3-1-10-dx-

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 \dots . t h a n k s

Answered by Ñï= last updated on 18/Mar/21

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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