高数∫1/（a^2-x^2)^2dx详细解答

Question

高数∫1/（a^2-x^2)^2dx详细解答

Answer 1

设x=atant t=arctan(x/a)
=1/a*∫cos^2t/cos2t *sec^2tdt=1/2a*∫1/cos2td2t
=ln |tan2t+sec2t| + C t=arctan(x/a)
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设 u=tant=x/a
tan2t=2u/1-u^2
sec2t=1/cos2x=1+u^2/1-u^2
tan2t+sec2t=(1+u)^2/(1-u^2)
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=2ln(1+x/a)-ln(1-(x/a)^2)+C
=2ln|a+x|-ln|a^2-x^2|+C