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Problem Let I ( x , a ) = ∫ d x x 2 + a 2 I(x,a)=\displaystyle \int \frac{dx}{x^{2}+a^{2}} I ( x , a ) = ∫ x 2 + a 2 d x a ≠ 0 a\neq 0 a = 0 ∂ I ∂ a \displaystyle \frac{\partial I}{\partial a} ∂ a ∂ I − 1 a 2 tan − 1 ( x a ) + x a ( x 2 + a 2 ) -\dfrac{1}{a^{2}}\tan^{-1}\!\left(\dfrac{x}{a}\right)+\dfrac{x}{a(x^{2}+a^{2})} − a 2 1 tan − 1 ( a x ) + a ( x 2 + a 2 ) x 1 a 2 tan − 1 ( x a ) − x a ( x 2 + a 2 ) \dfrac{1}{a^{2}}\tan^{-1}\!\left(\dfrac{x}{a}\right)-\dfrac{x}{a(x^{2}+a^{2})} a 2 1 tan − 1 ( a x ) − a ( x 2 + a 2 ) x − 1 a 2 tan − 1 ( x a ) − x a ( x 2 + a 2 ) -\dfrac{1}{a^{2}}\tan^{-1}\!\left(\dfrac{x}{a}\right)-\dfrac{x}{a(x^{2}+a^{2})} − a 2 1 tan − 1 ( a x ) − a ( x 2 + a 2 ) x − 1 a 2 tan − 1 ( x a ) -\dfrac{1}{a^{2}}\tan^{-1}\!\left(\dfrac{x}{a}\right) − a 2 1 tan − 1 ( a x ) Browse the problem library