Question

求下列几题打圈的题的积分

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Answer 1

1、原式=∫(0,π/2)dθ*∫(0,2)ln(1+r^2)rdr
=∫(0,π/2)dθ*(1/2)*[(1+r^2)ln(1+r^2)-r^2]|(0,2)
=∫(0,π/2)[(5/2)*ln5-2]dθ
=[(5/2)*ln5-2]θ|(0,π/2)
=(5ln5-4)*(π/4)
2、原式=∫(0,2π)dθ*∫(0,2π)sin(r^2)rdr
=∫(0,2π)dθ*(1/2)*[-cos(r^2)]|(0,2π)
=(1/2)*∫(0,2π)[1-cos(4π^2)]dθ
=(1/2)*[1-cos(4π^2)]θ|(0,2π)
=[1-cos(4π^2)]π
3、y=x转化为θ=π/4
y=√3x转化为θ=π/3
原式=∫(π/4,π/3)dθ*∫(√2,2)(1/r)rdr
=∫(π/4,π/3)(2-√2)dθ
=(2-√2)*(π/12)