Question

1 计算不定积分{arctanxdx 那个符号是积分符号哦 2 已知ln(x^2+y^2)=x^3y+sinx,求y..

Answer 1

1.原式=xarctanx-∫xdx/(1+x²) (应用分部积分法)

=xarctanx-1/2∫d(1+x²)/(1+x²)

=xarctanx-ln(1+x²)/2+C (C是积分常数)；

2.∵ln(x²+y²)=x³y+sinx ==>(2x+2yy')/(x²+y²)=x³y'+3x²y+cosx

==>[2y/(x²+y²)-x³]y'=3x²y+cosx-2x/(x²+y²)

==>y'=[3x²y+cosx-2x/(x²+y²)]/[2y/(x²+y²)-x³]

∴y'=[3x²y+cosx-2x/(x²+y²)]/[2y/(x²+y²)-x³].