歸約公式x2

1(a)
d/dx(x^n cosx-nx^n-1 sinx)
=x^n(-sinx)+ncosx(x^n-1)-nx^n-1cosx-n(n-1)sinx(x^n-2)
=-x^n sinx-n(n-1)x^n-2 sinx
(b)
∫[-x^n sinx-n(n-1)x^n-2 sinx]dx=x^n cosx-nx^n-1 sinx
∫[-x^n sinx]dx-∫[n(n-1)x^n-2 sinx]dx=x^n cosx-nx^n-1 sinx
∫[x^n sinx]dx=-x^n cosx+nx^n-1 sinx-n(n-1)Jn-2
Jn=-x^n cosx+nx^n-1 sinx-n(n-1)Jn-2

(c)
J4
=∫x^4sinx dx
=-x^4cosx+4x^3sinx-12∫x^2sinx dx
Since
∫x^2sinx dx
=-x^2cosx+2xsinx-2∫sinx dx
=-x^2cosx+2xsinx-2cosx
we have
J4
=-x^4cosx+4x^3sinx-12(-x^2cosx+2xsinx-2cosx)
=-x^4cosx+4x^3sinx+12x^2cosx-24xsinx+24cosx
2(a)
sin nx sinx+cosxcosnx
=cos(n-1)x
so sin nx sinx=cos(n-1)x-cosx cosnx
(b)
d/dx(cos^m x sin nx)
=ncos^mxcosnx-msinnxcos^m-1xsinx
=(m+n)cos^m x cos nx-mcos^m-1 x( cosxcos nx+sinnxsinx)
=(m+n)cos^m x cos nx -mcos^m-1 x cos(n-1)x
(c)
∫(m+n)cos^m x cos nx -mcos^m-1 x cos(n-1)x=cos^m x sin nx
∫(m+n)cos^m x cos nx dx-∫mcos^m-1 x cos(n-1)x dx=cos^m x sin nx
(m+n)∫cos^m x cos nx dx=cos^m x sin nx+m∫cos^m-1 x cos(n-1)x dx
∫cos^m x cos nx dx=1/(m+n)[cos^m x sin nx+m∫cos^m-1 x cos(n-1)x dx]
I m,n=1/(m+n) cos^m x sinnx+m/(m+n) Im-1,n-1
(d)
I3,2
=∫cos^3x cos2x dx
=(1/5) cos^3 x sin2x+(3/5) I2,1
I2,1
=∫cos^2x cosx dx
= (1/3) cos^2 x sinx+(2/3) I1,0
I1,0
=∫cosx dx
=-sinx
so
I2,1= (1/3) cos^2 x sinx-(2/3)sinx
I3,2
=(1/5) cos^3 x sin2x+(3/5) [(1/3) cos^2 x sinx-(2/3)sinx]
=(1/5) cos^3 x sin2x+(1/5) cos^2 x sinx-(2/5)sinx