∫x3sinxdx
Using the formula,∫Udv=UV−∫Vdu
U=x3dxdu=3x2du=3x2dx;dv=sinxdx;∫dv=∫sinxdx;V=−cosx
∴∫x3sinxdx=(x3)(−cosx)−∫(−cosx)(3x2dx)
∫x3sinxdx=−x3cosx−∫(−3x2cosxdx)
∫x3sinxdx=−x3cosx−(−3)∫x2cosxdx
∫x3sinxdx=−x3cosx+3∫x2cosxdx −−(i)
∫x2cosxdx=
U=x2dxdu=2xdu=2xdx;dv=cosxdx;∫dv=∫cosxdx;V=sinx
∴∫x2cosxdx=(x2)(sinx)−∫(sinx)(2xdx)
∫x2cosxdx=x2sinx−2∫xsinxdx −−−(ii)
∫xsinxdx=
U=xdxdu=1du=dx;dv=sinxdx;∫dv=∫sinxdx;V=−cosx
∴∫xsinxdx=(x)(−cosx)−∫(−cosx)(dx)
∫xsinxdx=−xcosx+∫cosxdx
∫xsinxdx=−xcosx+sinx
Substituting the value of ∫xsinxdx into equation (ii),
∫x2cosxdx=x2sinx−2∫xsinxdx
∫x2cosxdx=x2sinx−2(−xcosx+sinx)
∫x2cosxdx=x2sinx+2xcosx−2sinx
Substituting the value of ∫x2cosxdx into equation (i), we have
∫x3sinxdx=−x3cosx+3∫x2cosxdx
∫x3sinxdx=−x3cosx+3(x2sinx+2xcosx−2sinx)
∫x3sinxdx=−x3cosx+3x2sinx+6xcosx−6sinx+C
∴The integral of x3sinx=−x3cosx+3x2sinx+6xcosx−6sinx+C
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