How is the integral calculator finding this ∫ 0 a a 2 − x 2...

Lena Bell

Lena Bell

Answered

2022-07-01

How is the integral calculator finding this 0 a a 2 x 2 ( a 2 + x 2 ) 2 d x while I get an undefined expression?

Answer & Explanation

conveneau71

conveneau71

Expert

2022-07-02Added 17 answers

Alternatively, both cloudy's suggestion and the following method (the third step uses θ = arcsinh x a and integration by parts) also avoid having to divide by 0:
a 2 x 2 ( a 2 + x 2 ) 2 d x = 2 a 2 ( a 2 + x 2 ) 2 1 a 2 + x 2 d x = 2 a sech 3 ( arcsinh x a ) d ( arcsinh x a ) 1 a arctan x a = 1 a ( sech θ tanh θ + sech θ d θ ) 1 a arctan x a = 1 a ( sinh θ sech 2 θ + arctan x a ) 1 a arctan x a + C = 1 a ( x a × 1 1 + ( x a ) 2 ) + C = x a 2 + x 2 + C .
Alternatively, use θ = arctan x a or integrate by reduction formula.

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