∫cosx/(sin2x-6sinx+12)的∫cos/(sin^2-6sinx+12)dx

Question

∫cosx/(sin2x-6sinx+12)的
∫cos/(sin^2-6sinx+12)dx

Answer 1

dsinx = cosx *dx
cosx dx/(sin^2 x - 6sinx +12) = dsinx/(sin^2 x -6sinx +12)
令t=sinx有dt/(t^2-6t+12)= dt/((t-3)^2+3)
令y=t-3,则t=y+3,dt=dy有 dy/(y^2+3) = dy/((y/√3)^2+1)/3
令z=y/√3,则y=√3*z有dy=√3dz,所以 化为√3dz/(z^2+1)/3
而dx/(x^2+1)积分为arctanx
所以积分结果为arttanz *√3/3 = arctan(y/√3)*√3/3=arctan((t-3)/√3)*√3/3=arctan((sinx-3)/√3)*√3/3
最后再加上常数C就得到结果为
arctan((sinx-3)/√3)*√3/3 + C