Finding $\int\limits_1^\infty \frac{\sin^4(\log x)}{x^2 \log x} \mathrm{d}x$

The integral screams for a sub $x=e^u$; the result is

$$\int_0^{\infty} du \, e^{-u} \frac{\sin^4{u}}{u} $$

This is very computable by introducing a parameter and differentiating under the integral. In this case, consider

$$F(k) = \int_0^{\infty} du \, e^{-k u} \frac{\sin^4{u}}{u} $$

$$F'(k) = -\int_0^{\infty} du \, e^{-k u} \sin^4{u} $$

$F'(k)$ is relatively easy to compute using the fact that $\sin^4{u} = \frac{3}{8}-\frac12 \cos{2 u} + \frac18 \cos{4 u}$, and that

$$\int_0^{\infty} du \, e^{-k u} \cos{m u} = \frac{k}{k^2+m^2}$$

Thus

$$F'(k) = -\frac{3}{8 k} + \frac12 \frac{k}{k^2+4} - \frac18 \frac{k}{k^2+16} $$

and

$$F(k) = -\frac1{16} \log{\left [ \frac{k^6 (k^2+16)}{(k^2+4)^4} \right ]} + C$$

To evaluate $C$, we must consider $\lim_{k \to \infty} F(k)$ because $F(0)$ represents a non convergent integral. Because the limit is zero, we must have $C=0$. The integral we seek is then

$$F(1) = \frac1{16} \log{\frac{625}{17}}$$

I propose the following approach:

• Let $t=\dfrac1x$
• Let $u=\ln t$.
• Use the power-reduction formula for $\sin^4x$.
• Employ Euler's formula in conjunction with the linear properties of the integral.
• Express your new integral(s) in terms of $I(k)=\displaystyle\int_0^\infty e^{-kx}~dx$. You'll have $\displaystyle\int I(k)~dk$, where $I(k)$ will turn out to be a rational function in k. Solve it by using the usual methods.