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Question Number 142945 by Rankut last updated on 07/Jun/21

∫_0 ^(0.5) ((1+x^3 ))^(1/3) dx

∫00.51+x33dx

Answered by mindispower last updated on 07/Jun/21

(1+y)^a =1+Σ_(n≥1) ((Π_(k=0) ^(n−1) (a−k))/(n!))x^n (1+x^3 )^(1/3) =1+Σ_(n≥1) ((Π_(k=0) ^(n−1) (−1)^n (k−(1/3)))/(n!))x^n =(1/2)(1+Σ_(n≥1) (((k−(1/3))_n .n!)/((n+1)!)).(((((−1)/2))^n )/(n!))) =(1/2) _2 F_1 (−(1/3),1;2;[−(1/2)])

(1+y)a=1+∑n⩾1∏n−1k=0(a−k)n!xn(1+x3)13=1+∑n⩾1∏n−1k=0(−1)n(k−13)n!xn=12(1+∑n⩾1(k−13)n.n!(n+1)!.(−12)nn!)=122F1(−13,1;2;[−12])

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