Let f be a polynomial function such that for all real x f(x2+1)=x4+5 x2+3 then the premitive of f( x ) w.r.t. x, is

Question

Let f be a polynomial function such that for all real x f(x2+1)=x4+5 x2+3 then the premitive of f( x ) w.r.t. x, is (A) (x3/3)+(3 x2/2)-x+C (B) (x3/3)-(3 x2/2)+x+C (C) (x3/3)-(3 x2/2)-x+C (D) (x3/3)+(3 x2/2)+x+C

Tardigrade Admin · Accepted Answer

f(x2+1)=(x2+1)2+3 x2+2=(x2+1)2+3(x2+1)-1 text put x 2+1= t f ( t )= t 2+3 t -1 text now integrate ∴ f ( x )= x 2+3 x -1

Q. Let $f$ be a polynomial function such that for all real $x$ $f (x^{2} + 1) = x^{4} + 5 x^{2} + 3$ then the premitive of $f (x)$ w.r.t. $x$ , is

$\frac{x ^{3}}{3} + \frac{3 x ^{2}}{2} - x + C$

$\frac{x ^{3}}{3} - \frac{3 x ^{2}}{2} + x + C$

$\frac{x ^{3}}{3} - \frac{3 x ^{2}}{2} - x + C$

$\frac{x ^{3}}{3} + \frac{3 x ^{2}}{2} + x + C$

$f (x^{2} + 1) = (x^{2} + 1)^{2} + 3 x^{2} + 2 = (x^{2} + 1)^{2} + 3 (x^{2} + 1) - 1$
$put x^{2} + 1 = t$
$f (t) = t^{2} + 3 t - 1 now integrate$
$∴ f (x) = x^{2} + 3 x - 1$

Q. Let f be a polynomial function such that for all real x f(x2+1)=x4+5x2+3 then the premitive of f(x) w.r.t. x, is

3x3​+23x2​−x+C

3x3​−23x2​+x+C

3x3​−23x2​−x+C

3x3​+23x2​+x+C