Question

求（x^2）/(x^2+a^2) 从x=0到1的积分

Answer 1

积分:(x^2)/(x^2+a^2)dx

=积分:[(x^2+a^2)-a^2]/(x^2+a^2)dx

=积分:[1-a^2/(x^2+a^2)]dx

=x-a^2/a*arctanx/a+C

=x-aarctanx/a+C

所以

积分:(0,1)(x^2)/(x^2+a^2)dx

=1-aarctan1/a-(0-aarctan0)

=1+pai/4-arctan1/a