兩條f5math question ?10點

The solutions are as follows:

圖片參考：http://img714.imageshack.us/img714/9063/math0002.pdf


2011-06-10 22:39:56 補充：
http://img714.imageshack.us/img714/9063/math0002.pdf

Find following indefinite integrals
a.
∫(1/(3x+2)^1/2)dx
= (1/3)∫(3x+2)^(-1/2)d(3x+2)
= (1/6)(3x+2)^(1/2) + C
b.
∫(u(u-1)^1/2)dx
= u(u-1)^(1/2) x + C


2011-06-10 19:33:38 補充：
∫(u(u-1)^(1/2)du
= (2/3) ∫ u d(u-1)^(3/2)
= (2/3) [u(u-1)^(3/2) - ∫(u-1)^(3/2)du]
= (2/3)[u(u-1)^(3/2) - ∫(u-1)^(3/2)d(u-1)]
= (2/3)[u(u-1)^(3/2) - (2/5)(u-1)^(5/2) + C1]
= 2u(u-1)^(3/2) /3 - 4(u-1)^(5/2) /15 + C2 for C2 = 2C1/3