Let _ n ( n n^4+1 - 2 n (n^2+1 ) n^4+1 + n n^4+16 - 8 n (n^2+4 ) n^4+16 . .+ + n n^4+n^4 - 2 n n^2 (n^2+n^2 ) n^4+n^4 ) be k , using only the principal values of the inverse trigonometric functions. Then k ^2 is equal to ________

Step-by-step solution

aligned & _ r =1 ^ n n ^4+ r ^4 - 2 nr ^2 ( n ^2+ r ^2 ) n ^4+ r ^4 & _ r =1 ^ 1 n 1+ ( r n )^4 - 2 ( 1 n ) ( r n )^2 (1+ ( r n )^2 ) 1+ ( r n )^4 & ₀^1 dx 1+ x ^4 - 2 x ^2 dx (1+ x ^2 ) 1+ x ^4 & ₀^1 1- x ^2 (1+ x ^2 ) 1+ x ^4 d x & ₀^1 1 x ^2 -1 ( x + 1 x ) x ^2+ 1 x ^2 dx aligned aligned & - ₀^1 1- 1 x^2 (x+ 1 x ) (x+ 1 x )^2-2 d x & x+ 1 x =t 1- 1 x^2 d x=d t aligned aligned & - _ ^2 dt t t ^2-2 & - _ ^2 tdt t ^2 t ^2-2 & take t ^2-2= ^2 & tdt = d & - _ ^ 2 d ( ^2+2 ) & - _ ^ 2 d ^2+2 aligned aligned & . -1 2 ⁻