If x f(x)=3(f(x))2+2, then ∫ (2 x2-12 x f(x)+f(x)/(6 f(x)-x)(x2-f(x))2) d x is equal to

Question

If x f(x)=3(f(x))2+2, then ∫ (2 x2-12 x f(x)+f(x)/(6 f(x)-x)(x2-f(x))2) d x is equal to (A)  (1/x2-f(x))+C (B) (1/x2+f(x))+C (C) (1/x+f(x))+C (D) (1/x-f(x))+C

Tardigrade Admin · Accepted Answer

f(x)+x f prime(x)=6 f(x) ⋅ f prime(x) ⇒ f(x)=(6 f(x)-x) f prime(x) text Now I =∫ (2 x(x-6 f(x))+f(x)/(6 f(x)-x)(x2-f(x))2) d x -∫ (2 x(6 f(x)-x)-(6 f(x)-x) ⋅ f prime(x)/(6 f(x)-x)(x2-f(x))2) d x =-∫ (2 x-f prime(x)/(x2-f(x))2) d x put x2-f(x)=t (2 x-f prime(x) d x=d t. so I=(1/x2-f(x))+C

Q. If $x f (x) = 3 (f (x))^{2} + 2$ , then $\int \frac{2 x ^{2} - 12 x f ( x ) + f ( x )}{( 6 f ( x ) - x ) ( x ^{2} - f ( x ) ) ^{2}} d x$ is equal to

$\frac{1}{x ^{2} - f ( x )} + C$

$\frac{1}{x ^{2} + f ( x )} + C$

$\frac{1}{x + f ( x )} + C$

$\frac{1}{x - f ( x )} + C$

Q. If xf(x)=3(f(x))2+2, then ∫(6f(x)−x)(x2−f(x))22x2−12xf(x)+f(x)​dx is equal to

x2−f(x)1​+C

x2+f(x)1​+C

x+f(x)1​+C

x−f(x)1​+C

f(x)+xf′(x)=6f(x)⋅f′(x) ⇒f(x)=(6f(x)−x)f′(x) Now I =∫(6f(x)−x)(x2−f(x))22x(x−6f(x))+f(x)​dx −∫(6f(x)−x)(x2−f(x))22x(6f(x)−x)−(6f(x)−x)⋅f′(x)​dx =−∫(x2−f(x))22x−f′(x)​dx put x2−f(x)=t (2x−f′(x)dx=dt so I=x2−f(x)1​+C

Q. If $x f (x) = 3 (f (x))^{2} + 2$ , then $\int \frac{2 x ^{2} - 12 x f ( x ) + f ( x )}{( 6 f ( x ) - x ) ( x ^{2} - f ( x ) ) ^{2}} d x$ is equal to

$\frac{1}{x ^{2} - f ( x )} + C$

$\frac{1}{x ^{2} + f ( x )} + C$

$\frac{1}{x + f ( x )} + C$

$\frac{1}{x - f ( x )} + C$