dx 例12当a>0时,计算(1+8=2m且 解令x=2(t≥0),则有t=√x, 0.t=0. dx 故 1+t 2(1-,-)=2t-ln(1+) 1+t 2|√a-ln(+√a)
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
x dx 解令x=am,有√a2-x2= a cost,t=asth 且x=0,t=0; Va-x dx=2acost a cos tdt=a, cos2 tdt a 1+cos 2t (2+sin2)2=1zan2 4 注1由几何意义知,此定积分-xb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 −