计算这个积分
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计算这个积分
![](http://img.wesiedu.com/upload/b/38/b38c5116dc87fecd71919cfda70bed1d.jpg)
![](http://img.wesiedu.com/upload/b/38/b38c5116dc87fecd71919cfda70bed1d.jpg)
![计算这个积分](/uploads/image/z/18855901-37-1.jpg?t=%E8%AE%A1%E7%AE%97%E8%BF%99%E4%B8%AA%E7%A7%AF%E5%88%86)
∫[-π/4,π/4]x/(1+sinx)dx=∫[-π/4,π/4]x(1-sinx)/cos^2xdx
=∫[-π/4,π/4]xd(tanx-1/cosx)
=x(tanx-1/cosx)|[-π/4,π/4]-∫[-π/4,π/4](tanx-1/cosx)dx
=-π/√2+∫[-π/4,π/4](1-sinx)/cosxdx
=-π/√2+∫[-π/4,π/4]cosx/(1+sinx)dx
=-π/√2+ln(1+sinx)|[-π/4,π/4]
=-π/√2+2ln(√2+1)
=∫[-π/4,π/4]xd(tanx-1/cosx)
=x(tanx-1/cosx)|[-π/4,π/4]-∫[-π/4,π/4](tanx-1/cosx)dx
=-π/√2+∫[-π/4,π/4](1-sinx)/cosxdx
=-π/√2+∫[-π/4,π/4]cosx/(1+sinx)dx
=-π/√2+ln(1+sinx)|[-π/4,π/4]
=-π/√2+2ln(√2+1)