Question

利用欧拉积分求下列积分已知欧拉积分lnsinxdx=-pi*ln2/2,积分区间0-pi/2求定积分(arcsinx/x

Answer 1

Integrate[f(x),{x,0,1}],表示f(x)在积分区间0-1的值

Integrate[arcsin[x]/x,{x,0,1}]

=Integrate[arcsin[x]*(ln(x)'),{x,0,1}]

=ln(x)*arcsin(x)[0,1]-Integrate[(arcsin[x])'*ln(x),{x,0,1}]

=-Integrate[ln(x)/Sqrt(1-x^2),{x,0,1}] 令x=sin(y)

=-Integrate[ln(sin(y)),{y,0,Pi/2}]

=Pi*ln2/2