If 1-x^4 x^7 d x=f(x) 1-x^4 ^n+C , then (f(x))^n is equal to

Step-by-step solution

I= 1-x^4 x^7 d x Let x^2=u , then 2 x d x=d u I= 1-u^2 2 u^4 d u= 1 2 1-u^2 u^4 d u Let u= v , then d u= v d v aligned I & = 1 2 ^2 v ^4 v d v= 1 2 1 ^2 v ^4 v d v & = 1 2 ^2 v ^4 v d v aligned Let v=w , then ^2 v d v=d w aligned & I= 1 2 d w w^4 = 1 2 ( w⁻³ -3 )= -1 6 w^3 +C & = -1 6 ^3 v = - ( 1-u^2 )^3 6 u^3 =- 1 6 (1-x^4 )^ 3 / 2 x^6 +C & I= -1 6 (1-x^4 )^ 3 / 2 x^6 +C aligned Now, I=f(x) (1-x^4 )^ n / 2 +C on Comparing, we obtain gathered f(x)= -1 6 x^6 and n=3 (f(x))^n= (- 1 6 x^6 )^3= -1 216 x¹⁸ gathered