求∫sin^2(x)[sin^4(x)+ln(3+x)/(3-x)]dx在[-π/2,π/2]上的定积分

显示全部楼层 · 2011-9-15 11:11:07

设f(x)=sin2x{sin^4(x)+ln[(3+x)/(3-x)]}则f(x)=sin2x*sin^4(x)+sin2x*ln[(3+x)/(3-x)]令g(x)=sin2x*sin^4(x)=sin^6(x), h(x)=sin2x*ln[(3+x)/(3-x)]则f(x)=g(x)+h(x)f(-x)=g(-x)+h(-x)=sin^6(-x)+sin2(-x)*ln[(3-x)/(3+x)]=sin^6(x)-sin2x*ln[(3+x)/(3-x)]=g(x)-h(x)即g(x)为偶函数， h(x)为奇函数∴∴∫ (-π/2,π/2) g(x) dx=...

千问 · 2011-9-15 11:11:07

ln(3+x)/(3-x) 是奇函数，∫[-π/2,π/2]sin^2(x) * ln(3+x)/(3-x) dx = 0∫[-π/2,π/2] (sinx)^6 dx= 2 ∫[0,π/2] (sinx)^6 dx 有公式= 2 * 5*3/(6*4*2) * (π/2) = 5π/16...