Im=∫[0,π]x(sinx)^mdx =∫[0,π]x(sinx)^(m-1)d(-cosx) =-cosx*x(sinx)^(m-1)|[0,π]+∫[0,π]cosxdx(sinx)^(m-1) =∫[0,π]cosx*sinx^(m-1)dx+(m-1)∫[0,π]x(cosx)^2(sinx)^(m-2)dx =∫[0,π]sinx^(m-1)dsinx +(m-1)∫[0,π]x(1-sinx)^2(sinx)^(m-2)dx =(m-1)∫[0,π]x(sinx)^(m-2)dx+(m-1)Im =(m-1)Im-2+(m-1)Im Im=[(m-1)/m]Im-2 I0=∫[0,π]xdx=π^2/2 I1=∫[0,π]xsinxdx=∫[0,π]xd(-cosx)=x*(-cosx)|[0,π] +∫[0,π]cosxdx=π I2=∫[0,π]x(sinx^2)dx=∫[0,π]xsinxd(-cosx)=∫[0,π]cosxd(xsinx)=∫[0,π]sinxdsinx+∫[0,π]x(cosx)^2dx =∫[0,π]xdx-∫[0,π]xsinx^2dx 2∫[0,π]x(sinx)^2dx=π^2/2 I2=∫[0,π]x(sinx)^2dx=π^2/4 I3=[(3-1)/3]I1=(2/3)π I(2n)=[(1*3*5*..*(2n-1))/(2*4*6*..*2n)] (π^2/2) I(2n-1)=[(2*4*..*(2n-2))/(1*3*5*..*(2n-1))] π 會高數,純複製~