If α, β(βα) are the roots of f(x) ≡ a x2+b x+c=0, a ≠ 0 and f(x) is an even function and I=∫αβ (eff((f(x)/x-α))/f((f(x))x-α)+ef((f(x))x-β), then |I| is equal to

Question

If α, β(β>α) are the roots of f(x) ≡ a x2+b x+c=0, a ≠ 0 and f(x) is an even function and I=∫αβ (eff((f(x)/x-α))/f((f(x))x-α)+ef((f(x))x-β), then |I| is equal to (A) |(b/b)| (B)  mid (b/2 a) (C) (√b2-4 a c/|2 a|) (D) None of these

Tardigrade Admin · Accepted Answer

I=∫αβ (ef((a(x-α)(x-β)/x-α))/ef((a(x-α)(x-β))x-α)ef((a(x-α)(x-β))x-β) d x =∫/α)β (ef(a(x-β))/ef(a(x-β))+f(a(x-α))) d x ldots(1) =∫αβ (ef(a(α+β-x-β))/ef(a(α+β-x-β))+ef(a(α+β-x-α))) d x I=∫αβ (ef(a(x-α))/ef(a(x-α)+e∫(a(x-β))) d x ldots(2) 2 I=∫αβ d x ⇒ I=(|α-β|/2)=(√b2-4 a c/2|a|)

Q. If $α, β (β > α)$ are the roots of $f (x) \equiv a x^{2} + b x + c = 0, a \neq = 0$ and $f (x)$ is an even function and $I = \int_{α}^{β} \frac{e _{f}^{f (\frac{f ( x )}{x - α})}}{f ( \frac{f ( x )}{x - α} ) + e ^{f} ( \frac{f ( x )}{x - β} )}$ , then $∣ I ∣$ is equal to

$∣ ∣ \frac{b}{b} ∣ ∣$

$∣ \frac{b}{2 a}$

$\frac{b ^{2} - 4 a c}{∣2 a ∣}$

None of these

Q. If α,β(β>α) are the roots of f(x)≡ax2+bx+c=0,a=0 and f(x) is an even function and I=∫αβ​f(x−αf(x)​)+ef(x−βf(x)​)eff(x−αf(x)​)​​, then ∣I∣ is equal to

∣∣​bb​∣∣​

∣2ab​

∣2a∣b2−4ac​​

None of these

Q. If $α, β (β > α)$ are the roots of $f (x) \equiv a x^{2} + b x + c = 0, a \neq = 0$ and $f (x)$ is an even function and $I = \int_{α}^{β} \frac{e _{f}^{f (\frac{f ( x )}{x - α})}}{f ( \frac{f ( x )}{x - α} ) + e ^{f} ( \frac{f ( x )}{x - β} )}$ , then $∣ I ∣$ is equal to

$∣ ∣ \frac{b}{b} ∣ ∣$

$∣ \frac{b}{2 a}$

$\frac{b ^{2} - 4 a c}{∣2 a ∣}$