integration by substitution

Since u = 3x+ 2

du/dx = 3

So du = 3dx

∫3 (3x + 2)7 dx = ∫u7 du

= u8/8 + C

= (3x + 2)8/8 + C

2011-08-16 15:20:33 補充：
du, dx are the integration variable
du: Integrate w.r.t. u
dx: Integrate w.r.t. x

Consider that d(4^x)/dx = (4^x) ln 4, we have:

∫ 4^x dx = (4^x)/(ln 4) + C

2011-08-17 09:19:52 補充：
du means the "small change in quantity u"
Comparing to dx, if du/dx = f'(x), we have:

du = f'(x) dx

which means how are the "small change" du and dx are related.

e.g. if u = sin x, then we have du = cos x dx. So to find sin 31 without using calculator, we can use this relation by:

2011-08-17 09:19:57 補充：
u = 1/2 when x = π/6

With dx = π/180 (from 30 deg. to 31 deg.)

du = (cos π/6) x π/180