Question

Вычислить интеграл функции, №1

Answer 1

$aneq 0:;;;;x=a*tgt=>t=arctg dfrac{x}{a}=>dx=a*dfrac{1}{cos^2t}dt\ intdfrac{dx}{(x^2+a^2)^frac{3}{2}}=int dfrac{adt}{cos^2t*a^3*(1+tg^2t)^frac{3}{2}}=int dfrac{costdt}{a^2}=dfrac{1}{a^2}sint+C=dfrac{1}{a^2}dfrac{frac{x}{a}}{sqrt{(frac{x}{a})^2+1}}+C=dfrac{x}{a^2sqrt{x^2+a^2}}+C\ a=0:;;;;intdfrac{dx}{(x^2)^frac{3}{2}}=int dfrac{dx}{|x|^3}=int dfrac{|x|}{x}*dfrac{x}{|x|}*dfrac{1}{x^2|x|}dx=dfrac{x}{|x|}*int dfrac{1}{x^3}dx=-dfrac{1}{2x^2}dfrac{x}{|x|}+C=$$=-dfrac{1}{2x|x|}+C$