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The integral can be expressed in terms of Bessel and Struve functions, $$I(\xi)=\int_0^{\infty} dq\, \frac{J_0(q \xi)}{q+1}=\tfrac{1}{2} \pi \bigl(\pmb{H}_0(\xi)-Y_0(\xi)\bigr).$$ The small-$\xi$ behavior is $$I(\xi)=\xi-\ln \xi-\gamma_{\rm Euler} +\ln 2 +{\cal O}(\xi^2).$$ For large $\xi$ one has $$I(\xi)=1/\xi-\tfrac{1}{16}\sqrt\pi(\cos\xi+\sin\xi)+{\cal O}(\xi^{-2}).$$$$I(\xi)=1/\xi-\tfrac{1}{16}\sqrt\pi\xi^{-3/2}(\cos\xi+\sin\xi)+{\cal O}(\xi^{-2}).$$
The integral can be expressed in terms of Bessel and Struve functions, $$I(\xi)=\int_0^{\infty} dq\, \frac{J_0(q \xi)}{q+1}=\tfrac{1}{2} \pi \bigl(\pmb{H}_0(\xi)-Y_0(\xi)\bigr).$$ The small-$\xi$ behavior is $$I(\xi)=\xi-\ln \xi-\gamma_{\rm Euler} +\ln 2 +{\cal O}(\xi^2).$$ For large $\xi$ one has $$I(\xi)=1/\xi-\tfrac{1}{16}\sqrt\pi(\cos\xi+\sin\xi)+{\cal O}(\xi^{-2}).$$
The integral can be expressed in terms of Bessel and Struve functions, $$I(\xi)=\int_0^{\infty} dq\, \frac{J_0(q \xi)}{q+1}=\tfrac{1}{2} \pi \bigl(\pmb{H}_0(\xi)-Y_0(\xi)\bigr).$$ The small-$\xi$ behavior is $$I(\xi)=\xi-\ln \xi-\gamma_{\rm Euler} +\ln 2 +{\cal O}(\xi^2).$$ For large $\xi$ one has $$I(\xi)=1/\xi-\tfrac{1}{16}\sqrt\pi\xi^{-3/2}(\cos\xi+\sin\xi)+{\cal O}(\xi^{-2}).$$
The integral can be expressed in terms of Bessel and Struve functions, $$I(\xi)=\int_0^{\infty} dq\, \frac{J_0(q \xi)}{q+1}=\tfrac{1}{2} \pi \bigl(\pmb{H}_0(\xi)-Y_0(\xi)\bigr).$$ The small-$\xi$ behavior is $$I(\xi)=\xi-\ln \xi-\gamma_{\rm Euler} +\ln 2 +{\cal O}(\xi^2).$$ For large $\xi$ one has $$I(\xi)=1/\xi+{\cal O}(\xi^{-3/2}).$$$$I(\xi)=1/\xi-\tfrac{1}{16}\sqrt\pi(\cos\xi+\sin\xi)+{\cal O}(\xi^{-2}).$$
The integral can be expressed in terms of Bessel and Struve functions, $$I(\xi)=\int_0^{\infty} dq\, \frac{J_0(q \xi)}{q+1}=\tfrac{1}{2} \pi \bigl(\pmb{H}_0(\xi)-Y_0(\xi)\bigr).$$ The small-$\xi$ behavior is $$I(\xi)=\xi-\ln \xi-\gamma_{\rm Euler} +\ln 2 +{\cal O}(\xi^2).$$ For large $\xi$ one has $$I(\xi)=1/\xi+{\cal O}(\xi^{-3/2}).$$
The integral can be expressed in terms of Bessel and Struve functions, $$I(\xi)=\int_0^{\infty} dq\, \frac{J_0(q \xi)}{q+1}=\tfrac{1}{2} \pi \bigl(\pmb{H}_0(\xi)-Y_0(\xi)\bigr).$$ The small-$\xi$ behavior is $$I(\xi)=\xi-\ln \xi-\gamma_{\rm Euler} +\ln 2 +{\cal O}(\xi^2).$$ For large $\xi$ one has $$I(\xi)=1/\xi-\tfrac{1}{16}\sqrt\pi(\cos\xi+\sin\xi)+{\cal O}(\xi^{-2}).$$
The integral can be expressed in terms of Bessel and Struve functions, $$I(\xi)=\int_0^{\infty} dq\, \frac{J_0(q \xi)}{q+1}=\tfrac{1}{2} \pi \bigl(\pmb{H}_0(\xi)-Y_0(\xi)\bigr).$$ The small-$\xi$ behavior is $$I(\xi)=\xi-\ln \xi-\gamma_{\rm Euler} +\ln 2 +{\cal O}(\xi^2).$$ For large $\xi$ one has $$I(\xi)=1/\xi+{\cal O}(\xi^{-3/2}).$$
