x' In xdx例4求积分解u = Inx, x'dx =dy[ x' In xdxdxX116总结若被积函数是幂函数和对数函数或幂函数和反三角函数的乘积，就考虑设对数函数或反三角函数为u微积分经济数学
例4 求积分 3 x x x ln d . 解 u = ln x, 4 3 d d d , 4 x x x v = = 3 x x x ln d 1 1 4 3 ln d 4 4 = − x x x x . 16 1 ln 4 1 4 4 = x x − x + C 总结 若被积函数是幂函数和对数函数或幂 函数和反三角函数的乘积，就考虑设对数函 数或反三角函数为 u.
例5求积分sin(ln x)dx.解sin(Inx)dx= x sin(lnx) - [xd[sin(ln x)]J xcos(ln x).二 dx= xsin(lnx)- lx= xsin(Inx) - xcos(lnx) + ( xd[cos(Inx))sin(ln x)dx= x[sin(ln x) - cos(lnx)lsin(ln x)dx-[sin(In x) - cos(ln x)I + C.2经济数学微积分
例5 求积分 sin(ln )d . x x 解 sin(ln )d x x = − x x x x sin(ln ) d[sin(ln )] 1 x x x x x sin(ln ) cos(ln ) d x = − = − + x x x x x x sin(ln ) cos(ln ) d[cos(ln )] = − − x x x x x [sin(ln ) cos(ln )] sin(ln )d sin(ln )d x x [sin(ln ) cos(ln )] . 2 x = − + x x C