分部积分 ∫ x^3(1+x^2)^(1/2)

题目

分部积分 ∫ x^3(1+x^2)^(1/2)
是 ∫ x^3(1+x^2)^(1/2)dx

答案

其实这一道题,如果不用分部积分法,用代换法更有效,当然了,分部积分法也可以,具体如下：
∫ x^3(1+x^2)^(1/2)dx
=∫ x^2(1+x^2)^(1/2)×xdx=1/2∫ x^2(1+x^2)^(1/2)dx^2
=1/2∫ x^2(1+x^2)^(1/2)d(x^2+1)
=1/2∫ x^2×2/3d(x^2+1)^(3/2)
=1/3×x^2×(x^2+1)^(3/2)-1/3×∫ (x^2+1)^(3/2)dx^2
=1/3×x^2×(x^2+1)^(3/2)-1/3×∫ (x^2+1)^(3/2)d(x^2+1)
其实这一道题,如果不用分部积分法,用代换法更有效,当然了,分部积分法也可以,具体如下：
∫ x^3(1+x^2)^(1/2)dx
=∫ x^2(1+x^2)^(1/2)×xdx=1/2∫ x^2(1+x^2)^(1/2)dx^2
=1/2∫ x^2(1+x^2)^(1/2)d(x^2+1)
=1/2∫ x^2×2/3d(x^2+1)^(3/2)
=1/3×x^2×(x^2+1)^(3/2)-1/3×∫ (x^2+1)^(3/2)dx^2
=1/3×x^2×(x^2+1)^(3/2)-1/3×2/5×(x^2+1)^(5/2)
=1/3×(x^2+1)^(3/2)(3/5x^2+2/5)+C（C为积分常数）