求定积分,上限2派,下限0,根号下(1+cosx)de答案为4根号2

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求定积分,上限2派,下限0,根号下(1+cosx)de答案为4根号2
求定积分,上限2派,下限0,根号下(1+cosx)de答案为4根号2

求定积分,上限2派,下限0,根号下(1+cosx)de答案为4根号2
√(1+cosx)
=√[1+2cos^2(x/2)-1]
=√[2cos^2(x/2)]
=√2*cos(x/2)
∫[0,π/2]√(1+cosx)dx
=∫[0,π/2]√2*cos(x/2)dx
=2√2sin(x/2)[0,π/2]
=2√2*√2/2
=2

y=√(1+cosx)是对称函数,对称轴是x=π,
√(1+cosx)
=√[1+2cos^2(x/2)-1]
=√[2cos^2(x/2)]
=√2*cos(x/2).
∫[0,2π]√(1+cosx)dx
=4∫[0,π]√2*cos(x/2)dx/2
=-4√2sin(x/2)[0,π]
=-4√2*(-1)
=4√2.