intlimits^∞_0 {frac{dx}{0(x+sqrt{x^{2} +1} )^{3}} } , (a) 3/8 (b) 3/4 (c) 3/2 (d) 8/5

I = ∫^∞_0 dx/(x+√x^2+1)^3

Put x = taut ⇒dx = sec² + dt

 ⇒ I = π/2∫_0 sec²+dt/(taut+√ 1+tan²t)^3

= π/2∫_0 sec²+dt/(taut+sect)^3

= π/2∫_0 cost dt/(1+sint)^3

Let 1+sint = u

⇒cos t dt = du

⇒intlimits^2_1 {} , frac{du}{u^{3} }

intlimits^2_1 {u^{-3} } , du

= [u^-2/-2]^2_1 = -1/2 [1/u^2]^2_1

= -1/2 (1/4-1) = 3/8

Ans. option (a) 3/8