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How can I evaluate the integral of $j_1 ^2(x)\exp(-bx)/x$ from 0 to ∞?

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If you mean the Bessel function $J_1(x)$:

Assuming[b > 0,
  Integrate[BesselJ[1, x]^2*Exp[-b x]/x, {x, 0, ∞}]]

(π + b^2 EllipticE[-4/b^2] - (4 + b^2) EllipticK[-4/b^2])/(2π)

If you mean the spherical Bessel function $j_1(x)$, it's a bit harder:

J[b_, x_] = Integrate[SphericalBesselJ[1, x]^2*Exp[-b x]/x, x];
Assuming[b > 0,
  Limit[J[b, x], x -> ∞] - Limit[J[b, x], x -> 0, Direction -> "FromAbove"] // FullSimplify]

1/96 (-4 (-6 + b^2) - 32 b ArcTan[2/b] + b^2 (12 + b^2) (-2 Log[b] + Log[-2 I + b] + Log[2 I + b]))

The last line can be simplified to

1/96 (-4 (-6 + b^2) - 32 b ArcTan[2/b] + b^2 (12 + b^2) Log[1 + 4/b^2])

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  • $\begingroup$ Thanks for the answer. But can you please provide me the steps for the solution of the integration. $\endgroup$ – Parveen Bano Jun 28 '19 at 6:50
  • $\begingroup$ @ParveenBano you're on the wrong forum for that (this forum is about the Mathematica software). Maybe ask at the Mathematics StackExchange? $\endgroup$ – Roman Jun 28 '19 at 6:56
  • $\begingroup$ Thanks for the suggestion $\endgroup$ – Parveen Bano Jun 28 '19 at 7:42

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