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# Given ∫ X 0 1 X 2 + a 2 D X = 1 a Tan − 1 ( X a ) Using Duis Find the Value of ∫ X 0 1 X 2 + a 2 - Applied Mathematics 2

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Sum

Given int_0^x 1/(x^2+a^2) dx=1/atan^(-1)(x/a)using DUIS find the value of int_0^x 1/(x^2+a^2)

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#### Solution

int_0^x 1/(x^2+a^2) dx=1/atan^(-1)(x/a)

Differentiate w.r.t a , taking ‘a’ as parameter

d/(da)int_0^x 1/(x^2+a^2) dx=d/(da)[1/atan^(-1)(x/a)]

Applying D.U.I.S rule,

D.U.I.S rule says that if function and its partial derivative is continuous then we can apply differential operator in the integral operator by converting it into partial derivative taking one parameter fro function.

int_0^xdel/(dela) 1/(x^2+a^2) dx=-1/atan^(-1)(x/a)xx1/a+(-x)/(a(x^2+a^2)

int_0^x(2a^2)/(x^2+a^2) dx=-1/atan^(-1)(x/a)xx1/a+(-x)/(a(x^2+a^2)

int_0^x(dx)/(x^2+a^2)^2 dx=-1/(2a^3)tan^(-1)  x/a+x/(2a^2(x^2+a^2)

Concept: Method of Variation of Parameters
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