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∫（1+x^3)^(3/2)*x^5dx
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∫（1+x^3)^(3/2)*x^5dx
▼优质解答
答案和解析
u = 1+x³,du = 3x² dx
∫ (1+x³)^(3/2) * x^5 dx
= (1/3)∫ u^(3/2) * (u-1) du
= (1/3)∫ [u^(5/2) - u^(3/2)] du
= (1/3) * [u^(5/2+1) / (5/2+1) - u^(3/2+1) / (3/2+1)] + C
= 2/21*(1+x³)^(7/2) - 2/15*(1+x³)^(5/2) + C
= (2/105)(5x³-2)(1+x³)^(5/2) + C
∫ (1+x³)^(3/2) * x^5 dx
= (1/3)∫ u^(3/2) * (u-1) du
= (1/3)∫ [u^(5/2) - u^(3/2)] du
= (1/3) * [u^(5/2+1) / (5/2+1) - u^(3/2+1) / (3/2+1)] + C
= 2/21*(1+x³)^(7/2) - 2/15*(1+x³)^(5/2) + C
= (2/105)(5x³-2)(1+x³)^(5/2) + C
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