uUUkxdx,P,()P.(P.()sinaxdx,cosaxdxDe其中k,a为常数P,(x)为n次多项式
P (x)sinaxdx, P (x)cosaxdx, P (x)e dx, kx n n n 其中k,a为常数, u u u Pn (x)为n次多项式
[udv = uv - [ vdu例3求xarctanxdx.u=x,dv=arctanxdxdu=dx, v:解 u=arctanx, dv = xdx.21xdudx.21+ xxarctanxdxdxarctan221+.1)dxarctan x221+x1(x -arctanx) +Carctanx -22
例3 求 arctan d . x x x 解 u = arctan x , 2 2 x v = xarctanxdx = x − x arctan 2 2 = x − x arctan 2 2 x x x C x = − ( − arctan ) + 2 1 arctan 2 2 d , 1 1 d 2 x x u + = dv = xdx udv uv vdu = − x x x d 1 1 2 2 2 + x x )d 1 1 (1 2 1 2 + − u = x, dv = arctan xdx du = dx, v =
-udv = uv-vduI x3 In xdx.例4 求dv = x'dx解 u=lnx,化简型xdu=二dx4x1x31Inxdxdx411C16X
例 4 求 ln d . 3 x x x 解 u = ln x , d v x dx 3 = x ln x dx 3 = x ln x − 41 4 = x x − x + C 4 4 161 ln 41 44 x x v = x u d1 d = 化简型 x dx 41 3 u d v uv vd u = −