∫ X²√x³+1 dx
use u-substitution
u = x³+1
du = 2x² dx
∫ X²√x³+1 dx =
1/2 ∫ du √u =
1/2 ∫ u^(1/2) du =
1/2 [u^(3/2)] (2/3) + C =
(1/3) [u^(3/2)] + C =
(1/3) [(x³+1)^(3/2)] + C
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Verified answer
∫ X²√x³+1 dx
use u-substitution
u = x³+1
du = 2x² dx
∫ X²√x³+1 dx =
1/2 ∫ du √u =
1/2 ∫ u^(1/2) du =
1/2 [u^(3/2)] (2/3) + C =
(1/3) [u^(3/2)] + C =
(1/3) [(x³+1)^(3/2)] + C