_ x [ (1+ 1 n^3 )^ 1 n^3 (1+ 8 n^3 )^ 4 n^3 (1+ 27 n^3 )^ 9 n^3 . .(2)^ 1 n ]=

Step-by-step solution

aligned & I = _ x [ (1+ 1 n^3 )^ 1 n^3 (1+ 8 n^3 )^ 4 n^3 (1+ 27 n^3 )^ 9 n^3 (2)^ 1 n ] & I = _ n [ (1+ 1 n^3 )^ 1 n^3 + (1+ 8 n^3 )^ 4 n^3 + + (2)^ 1 n ] & I = _ n 1 n^3 (1+ 1 n^3 )+ 4 n^3 (1+ 8 n^3 )+ + 1 n (2) & I = _ n _ r=1 ^n r^2 n^3 (1+ r^3 n^3 )= _ n 1 n _ r=1 ^n r^2 n^2 (1+ r^3 n^3 ) & I = ₀^1 x^2 (1+x^3 ) d x aligned Let 1+x^3=t 3 x^2 d x=d t aligned & I = 1 3 ₁^2 t d t I = 1 3 [t t-t]₁^2 & I = 1 3 [2 2-2+1] I =e^ ( 2 2-1 3 ) aligned