displaystyle∫0∞(dx/(x+√x2+1)3 )=

Question

displaystyle∫0∞(dx/(x+√x2+1)3 )= (A) (3/8) (B) (1/8) (C) -(3/8) (D) -(1/8)

Tardigrade Admin · Accepted Answer

Put x = tan θ ∴ dx=sec2 θ dθ When x = 0, θ=0, x=∞, θ=π /2 ∴ given integral =∫ limits0π /2 (cos θ dθ/(1+sin θ)3)=|((1+sin θ)-2/-2)|0π /2 =-(1/8)+(1/2)=(3/8)

Q. $\int_{0}^{\infty} \frac{d x}{( x + x ^{2} + 1 ) ^{3}} =$

$\frac{3}{8}$

$\frac{1}{8}$

$- \frac{3}{8}$

$- \frac{1}{8}$

Put $x = t an θ ∴ d x = se c^{2} θ d θ$
When $x = 0, θ = 0, x = \infty, θ = π /2$
$∴$ given integral
$= 0 \int π /2 \frac{cos θ d θ}{( 1 + s in θ ) ^{3}} = ∣ ∣ \frac{( 1 + s in θ ) ^{- 2}}{- 2} ∣ ∣_{0}^{π /2}$
$= - \frac{1}{8} + \frac{1}{2} = \frac{3}{8}$

Q. ∫0∞​(x+x2+1​)3dx​=

83​

81​

−83​

−81​