Question

求不定积分，第三个和第四个

见下图，谢谢！

Answer 1

这是我理解的，你自己看一下吧，兄弟。

Answer 2

3. 设 u = (1+x)^(1/3) , 则 x = u^3-1, dx = 3u^2du
I = ∫ 3u^2du/(1+u) = 3 ∫(u^2+u-u-1+1)du/(u+1)
= 3 ∫[u-1+1/(u+1)]du = 3[u^2/2 - u + ln(u+1)] + C
= (3/2)(1+x)^(2/3) - 3(1+x)^(1/3) + 3ln[1+(1+x)^(1/3)] + C
4. 设 u = √(2-5x), 则 x = (1/5)(2-u^2), dx = (-2/5)udu
I = ∫ (1/5)(2-u^2)u(-2/5)udu = (-2/25)∫ (2u^2-u^4)du
= (-2/25)(2u^3/3-u^5/5) + C
= (-4/75)(2x-5)^(3/2) + (1/125)(2x-5)^(5/2) + C