∫ [ ( sin x cos x ) / ( sin⁴ x + cos⁴ x ) ] dx
= (1/2) ∫ { [ ( 2 sin x cos x ) / cos⁴ x ] / [ ( sin⁴ x + cos⁴ x ) / cos⁴ x ] } dx
= (1/2) ∫ { [ 2 ( sin x / cos x ) ( 1/ cos² x ) ] / [ ( sin⁴ x / cos⁴ x ) + 1 ] } dx
= (1/2) ∫ [ ( 2 tan x sec² x ) / ( tan⁴ x + 1 ) ] dx
= (1/2) ∫ { 1 / [ 1 + ( tan² x )² ] } • 2 ( tan x ) ( sec² x ) dx
= (1/2) ∫ [ 1 / ( 1 + u² ) ] du, .......... u = tan² x, du = 2 ( tan x ) ( sec² x ) dx
= (1/2)· tanֿ¹ ( u ) + C
= (1/2)· tanֿ¹ ( tan² x ) + C