∫0∞ (d x/(x2+a2)(x2+b2)) is

Question

∫0∞ (d x/(x2+a2)(x2+b2)) is (A) (π a b/a+b) (B) (π/2(a+b)) (C) (π/2 ab ( a + b )) (D) (π(a+b)/2 a b)

Tardigrade Admin · Accepted Answer

∫0∞ (d x/(x2+a2)(x2+b2)) =(1/b2-a2) ∫0∞ ((x2+b2)-(x2+a2)/(x2+a2)(x2+b2)) =(1/b2-a2) ∫0∞[(1/x2+a2)-(1/x2+b2)] d x =(1/b2-a2)[(1/a) tan -1 (x/a)-(1/b) tan -1 (x/b)]0∞ =(1/b2-a2)[(π/2 a)-(π/2 b)]=(π/2 a b(a+b))

Q. $\int_{0}^{\infty} \frac{d x}{( x ^{2} + a ^{2} ) ( x ^{2} + b ^{2} )}$ is

$\frac{πab}{a + b}$

$\frac{π}{2 ( a + b )}$

$\frac{π}{2 ab ( a + b )}$

$\frac{π ( a + b )}{2 ab}$

Q. ∫0∞​(x2+a2)(x2+b2)dx​ is

a+bπab​

2(a+b)π​

2ab(a+b)π​

2abπ(a+b)​

∫0∞​(x2+a2)(x2+b2)dx​ =b2−a21​∫0∞​(x2+a2)(x2+b2)(x2+b2)−(x2+a2)​ =b2−a21​∫0∞​[x2+a21​−x2+b21​]dx =b2−a21​[a1​tan−1ax​−b1​tan−1bx​]0∞​ =b2−a21​[2aπ​−2bπ​]=2ab(a+b)π​

Q. $\int_{0}^{\infty} \frac{d x}{( x ^{2} + a ^{2} ) ( x ^{2} + b ^{2} )}$ is

$\frac{πab}{a + b}$

$\frac{π}{2 ( a + b )}$

$\frac{π}{2 ab ( a + b )}$

$\frac{π ( a + b )}{2 ab}$