xdx/cos(π/4-x)cosx
求:∫_0^(π/4)x/cosx^2求上限π/4,下限0,被积表达式x/cosx^2dx,答案(π-2ln2)/4...
求:∫_0^(π/4)x/cosx^2 求上限π/4,下限0,被积表达式x/cosx^2 dx,答案(π-2ln2)/4
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用分部积分法,
原式=∫[0,π/4] x*(secx)^2dx
=∫[0,π/4] x d(tanx)
=(x*tanx)[0,π/4]-∫[0,π/4]tanxdx
=π/4+∫[0,π/4]d(cosx)/cosx
=π/4+ln|cosx|[0,π/4]
=π/4+[ln(√2/2)-ln1]
=π/4+(1/2)ln2-ln2
=π/4-(1/2)ln2
=(π-2ln2)/4.
原式=∫[0,π/4] x*(secx)^2dx
=∫[0,π/4] x d(tanx)
=(x*tanx)[0,π/4]-∫[0,π/4]tanxdx
=π/4+∫[0,π/4]d(cosx)/cosx
=π/4+ln|cosx|[0,π/4]
=π/4+[ln(√2/2)-ln1]
=π/4+(1/2)ln2-ln2
=π/4-(1/2)ln2
=(π-2ln2)/4.
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