一、回归变换法：

 A=∫(0,π/2)[ln(sinx)]dx=2∫(0,π/4)[ln(2sinycosy)]dy

   =(π/2)ln2+2∫(0,π/4)[ln(siny)]dy+2∫(0,π/4)[ln(cosy)]dy

   =(π/2)ln2+2∫(0,π/4)[ln(siny)]dy+2∫(π/4,π/2)[ln(siny)]dy

   =(π/2)ln2+2A.

∴  A=(-π/2)ln2.

二、参数求导法：

F(t)=∫(0,1)[(x^t-1)/(lnx)]dx,

∵   d F(t)/dt =∫(0,1)x^tdx=1/(1+t)，F(0)=0，

∴   F(t)=∫(0,t)[1/(1+t)]d t =ln(1+t).

三、双重积分法：

（1）[∫(0,∞)e^{-x^2}dx]²=∫(0,∞)e^{-x^2}dx*∫(0,∞)e^{-y^2}dy

       =∫(0,∞)∫(0,∞)e^{-x^2-y^2}dxdy（令x=r*cosθ，y=r*sinθ）

      =∫(0,π/2)∫(0,∞)e^{-r^2}rdrdθ

      =π/4.

      ∴  ∫(0,∞)e^{-x^2}dx=(√π)/2.

（2）F(t)=∫(0,1)[(x^t-1)/(lnx)]dx=∫(0,1)∫(0,t)x^ydydx

              =∫(0,t)∫(0,1)x^ydxdy=∫(0,t)[1/(1+y)]dy

              =ln(1+t)

（3） ∫(0,∞)[(1/x)sinx]dx=∫(0,∞)sinx∫(0,∞)e^-xydydx

         =∫(0,∞)∫(0,∞)e^-xysinxdxdy=∫(0,∞)[1/(1+y²)]dy

         =π/2

四、级数变换法：

∫(0,∞)e^{-(ax)^2}ch(2bx)dx=∫(0,∞)e^{-(ax)^2}[∑(n=0…∞)(2bx)²ⁿ/(2n)!]dx

=∑(n=0…∞){[(2b)²ⁿ/(2n)!]∫(0,∞)e^{-(ax)^2}x²ⁿdx}（令(ax)²=y，a＞0）

=[1/(2a)]∑(n=0…∞){[(2b/a)²ⁿ/(2n)!]∫(0,∞)e^-yy^n-1/2dy}

=[1/(2a)]∑(n=0…∞){[(2b/a)²ⁿ/(2n)!]Γ(n+1/2)}

=[(√π)/(2a)]∑(n=0…∞){[(2b/a)²ⁿ/(2n)!][(2n-1)!!/2ⁿ]}

=[(√π)/(2a)]∑(n=0…∞)[(b/a)²ⁿ/n!]

=[(√π)/(2a)]e^{(b/a)^2}

五、复变留数法：（略）