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

Question

Accepted Answer

Correct answer: B. (b) Let, I= 1-x^4x^7 dx Take x^2=u, then 2 xdx= du I= 1-u^22u^4 du= 12 1-u^2u^4du Take u= sin ⁡v, then du= cos ⁡ vdv I= 12 cos^2 ⁡v sin^4 ⁡v dv= 12 cos^4 ⁡v cos^2 ⁡v sin^4v dv= 12 1 cos^2 ⁡v tan^4 ⁡v dv= 12 sec^2 ⁡v tan^4 ⁡v dv Take tan ⁡v=w, then sec^2 ⁡ vdv= dw I= 12 dww^4= 12 ( w^-3-3 )= -16w^3+C= -16 tan^3 ⁡v= - ( 1-u^2 )^36u^3=- 16 (1-x^4)^3/2x^6+C Now, on comparing with I=f (x )(1-x^4)n+C, we get f (x )= -16x^6 and n=3 (f (x ) )^n=(- 16x^6)^3= -1216x^18

$If \int \frac{1 - x ^{4}}{x ^{7}} dx = f (x) (1 - x^{4})^{n} + C, then (f (x))^{n} is equal to$

Solution

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