The integral\

\\\\int (x^(3))/((81-x^(2))^((5)/(2)))dx

\ can be reduced to the form\

k\\\\int sin^(m)(\\\\theta )cos^(n)(\\\\theta )d\\\\theta

\ with the substitution

x(\\\\theta )=

\ so that, in terms of

\\\\theta

,\

\\\\sqrt(81-x^(2))=,>=0,\ k=,m= and n=

\ The substitution

u(\\\\theta )=

reduces the integral to

\\\\int R(u)du

,\ where the rational function\

R(u)=

\ The partial fraction expansion of

R(u)

is\ The integral

\\\\int R(u)du=

\ from which we see that\

\\\\int (x^(3))/((81-x^(2))^((5)/(2)))d

The integral (81x2)25x3dx can be reduced to the form ksinm()cosn()d with the substitution x()= so that, in terms of , 81x2=k=0,,m=andn= The substitution u()= reduces the integral to R(u)du, where the rational function R(u)= The partial fraction expansion of R(u)is The integral R(u)du= from which we see that (81x2)25x3dx=