∫lnsin2xdx(0~π/4) (表示从0到π/4的定积分)
=∫ln(2sinx cosx)dx(0~π/4)
=π/4*ln2+∫lnsinxdx(0~π/4)+∫lncosxdx(0~π/4)
=π/4*ln2+∫lnsinxdx(0~π/4)+∫lnsinxdx(π/4~π/2) (对最后一个积分换元)
=π/4*ln2+∫lnsinxdx(0~π/2)
=π/4*ln2+2∫lnsin2xdx(0~π/4) (换元)
由第一个式子与最后一个式子相等即得
∫lnsin2xdx(0~π/4)=-π/4*ln2