例12当a>0时,计算()a 1+ 解令x=t2(t≥0),则有t=√x,dkx=2lh,且 x=0,t=0,x=a,t=√a 故 o1+√x 01+t a (1-,)t=2[t-ln(1+) 1+ 2va-In(1+va) 挂主接
6 例12 当 a > 0时, 计算 0 (1) 1 a dx + x 2 解 令 则有 且 ( 0), , 2 , x t t t x dx tdt = = = 0 1 2 (1 ) 2[ ln(1 )] 1 0 a a dt t t t = − = − + + x t x a t a = = = = 0, 0, , 0 0 2 1 1 a a dx t dt x t = + + 故 = − + 2[ ln(1 )] a a
(2)VG2-y2 解令x= asin t,有√a2-x2= a cos t,d= a cos tdt 且x=0,t=0; dx=2 acost. a cos tdt=a cos tdt 0 0 1+cos 2t (2t +sin 2t) 4 注1由几何意义知,此定积分a2-x即为圆
7 2 2 解 令 有 sin , cos , cos x a t a x a t dx a tdt = − = = 0, 0; , 2 x t x a t 且 = = = = 2 2 0 (2) a a x dx − 2 2 2 2 0 0 a t a tdt a tdt cos cos cos = = 2 2 0a a x dx − 2 20 1 cos 2 2 t a dt + = 2 2 1 2 (2 sin 2 ) 4 4 0 a t t a = + = 注1 由几何意义知, 此定积分 即为圆 2 2 0a a x dx −