aligned & d x (2 x+ x)^4 =A(1+ x)⁻⁵ & +B(1+ x)⁻⁶+C(1+ x)⁻⁷+k, then & A+B+C= aligned

Step-by-step solution

aligned & I= d x (2 x+ x)^4 = ^4 x (2 x+ ^2 x )^4 d x & = (1+ ^2 x ) (1+ x)^8 ^2 x d x aligned Let x=t ^2 x d x=d t So, aligned & I= 1+t^2 (1+t)^8 d t= (1+t)^2-2 t (1+t)^8 d t &= d t (1+t)^6 -2 1+t-1 (1+t)^8 d t &= d t (1+t)^6 -2 d t (1+t)^7 +2 d t (1+t)^8 &=- 1 5 (1+t)⁻⁵+ 2 6 (1+t)⁻⁶- 2 7 (1+t)⁻⁷+K &= -1 5 (1+ x)⁻⁵+ 1 3 (1+ x)⁻⁶ &- 2 7 (1+ x)⁻⁷+k aligned So, A= -1 5 , B= 1 3 and C=- 2 7 A+B+C=- 16 105 .