DLMF:10.22.E9 (Q3383)

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DLMF:10.22.E9
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    ∫ 0 x J 2 ⁢ n ⁡ ( t ) ⁢ d t = ∫ 0 x J 0 ⁡ ( t ) ⁢ d t - 2 ⁢ ∑ k = 0 n - 1 J 2 ⁢ k + 1 ⁡ ( x ) , ∫ 0 x J 2 ⁢ n + 1 ⁡ ( t ) ⁢ d t = 1 - J 0 ⁡ ( x ) - 2 ⁢ ∑ k = 1 n J 2 ⁢ k ⁡ ( x ) , formulae-sequence superscript subscript 0 𝑥 Bessel-J 2 𝑛 𝑡 𝑡 superscript subscript 0 𝑥 Bessel-J 0 𝑡 𝑡 2 superscript subscript 𝑘 0 𝑛 1 Bessel-J 2 𝑘 1 𝑥 superscript subscript 0 𝑥 Bessel-J 2 𝑛 1 𝑡 𝑡 1 Bessel-J 0 𝑥 2 superscript subscript 𝑘 1 𝑛 Bessel-J 2 𝑘 𝑥 {\displaystyle{\displaystyle\int_{0}^{x}J_{2n}\left(t\right)\mathrm{d}t=\int_{% 0}^{x}J_{0}\left(t\right)\mathrm{d}t-2\sum_{k=0}^{n-1}J_{2k+1}\left(x\right),% \quad\int_{0}^{x}J_{2n+1}\left(t\right)\mathrm{d}t=1-J_{0}\left(x\right)-2\sum% _{k=1}^{n}J_{2k}\left(x\right),}}
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    DLMF:10.22.E9
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