X的五次方*sinx的不定积分

X的五次方*sinx的不定积分
X的五次方*sinx的不定积分

X的五次方*sinx的不定积分
∫x^5*sinxdx
=-∫x^5dcosx
=∫cosxdx^5-x^5cosx
=5∫x^4cosxdx-x^5cosx
=5∫x^4dsinx-x^5cosx
=5(x^4sinx-∫sinxdx^4)-x^5cosx
=-20∫x^3sinxdx+5x^4sinx-x^5cosx
=20∫x^3dcosx+5x^4sinx-x^5cosx
=-20∫cosxdx^3+20x^3cosx+5x^4sinx-x^5cosx
=-60∫x^2dsinx+20x^3cosx+5x^4sinx-x^5cosx
=60∫sinxdx^2-60x^2sinx+20x^3cosx+5x^4sinx-x^5cosx
=120∫xsinxdx-60x^2sinx+20x^3cosx+5x^4sinx-x^5cosx
=-120∫xdcosx-60x^2sinx+20x^3cosx+5x^4sinx-x^5cosx
=120∫cosxdx-120xcosx-60x^2sinx+20x^3cosx+5x^4sinx-x^5cosx
=120sinx-120xcosx-60x^2sinx+20x^3cosx+5x^4sinx-x^5cosx+c
=(120-60x^2+5x^4)sinx-(120x-20x^3+x^5)cosx+c