I have solved the Problem using trigo. substn., but the same
can be done without it, as shown below :
#I=int-x^3/sqrt(9+9x^2)dx#,
#=-1/3int(x^2*x)/sqrt(x^2+1)dx#,
#=-1/3int[{(x^2+1)-1}x]/sqrt(x^2+1)dx#,
#=-1/3int{(x^2+1)/sqrt(x^2+1)-1/sqrt(x^2+1)}xdx#,
#=-1/3int{sqrt(x^2+1)-1/sqrt(x^2+1)}xdx#,
#=-1/3*1/2int{sqrt(x^2+1)-1/sqrt(x^2+1)}(2x)dx#,
#=-1/6int{(x^2+1)^(1/2)-(x^2+1)^(-1/2)}d(x^2+1)#,
#=-1/6{(x^2+1)^(1/2+1)/(1/2+1)-(x^2+1)^(-1/2+1)/(-1/2+1)}#,
#=-1/6{2/3(x^2+1)^(3/2)-2(x^2+1)^(1/2)}#,
#=-1/6*2/3(x^2+1)^(1/2){(x^2+1)^(3/2-1/2)-3(x^2+1)^(1/2-1/2)}#,
#=-1/9(x^2+1)^(1/2){(x^2+1)-3}#,
#=-1/9(x^2-2)sqrt(x^2+1)+C#, as before!