DUIS: ∫ e^(−x) sin bx / x dx

Method

Differentiation under the integral sign (Leibnitz rule): if I(b) = ∫₀^∞ f(x, b) dx, with limits independent of b, then dI/db = ∫₀^∞ ∂f/∂b dx.

Solution

Let I(b) = ∫₀^∞ e^(−x) (sin bx)/x dx.

Differentiate with respect to b:

dI/db = ∫₀^∞ e^(−x) cos bx dx

Use the standard result ∫₀^∞ e^(−ax) cos bx dx = a/(a² + b²) with a = 1:

dI/db = 1/(1 + b²)

Integrate with respect to b:

I(b) = tan⁻¹ b + C

When b = 0, sin bx = 0, so I(0) = 0. Since tan⁻¹ 0 = 0, this gives C = 0.

Answer: ∫₀^∞ e^(−x) sin bx / x dx = tan⁻¹ b.