Question

一道高数题,答案是（根号2）*π

Answer 1

原式是偶函数

所以

原式=2∫(0,+∞)(x²＋1)/(x^4+1)dx

=2∫(0,+∞)(1＋1/x²)/(x²+1/x²)dx

=2∫(0,+∞)1/(x²+1/x²) d(x-1/x)

=2∫(0,+∞)1/(x²+1/x²-2+2) d(x-1/x)

=2∫(0,+∞)1/[(x-1/x)²＋2] d(x-1/x)

=2/√2 arctan(x-1/x)/√2 |（0,+∞）

=√2 (π/2-(-π/2))

=√2 π