Euler-Mascheroni Constant as Difference of Integrals involving Cosine

Theorem

$\ds \int_0^1 \frac {1 - \cos x} x \rd x - \int_1^\infty \frac {\cos x} x \rd x = \gamma$

where $\gamma$ is the Euler-Mascheroni Constant.

Proof

From Characterization of Cosine Integral Function, we have:

$\ds \int_t^\infty \frac {\cos x} x \rd x = -\gamma - \ln t + \int_0^t \frac {1 - \cos x} x \rd x$

for all real $t > 0$.

Rearranging:

$\ds \int_0^t \frac {1 - \cos x} x \rd x - \int_t^\infty \frac {\cos x} x \rd x = \gamma + \ln t$

Setting $t = 1$ gives the result, from Natural Logarithm of 1 is 0.

$\blacksquare$

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 15$: Definite Integrals involving Trigonometric Functions: $15.65$