Find the definite integral of:
$$\int_0^\infty \frac{\sin^4(7x)-\sin^4(5x)}{x} \ \mathrm d x$$
This question is from a Belarusian mathematical olympiad. This is from the topic of definite integrals, but I can't think of anything.
I tried using the Newton-Leibniz rule which we use generally to solve such questions. But it did not work. Please help.
Best Answer
Note $\sin^4(kx )= \frac38 +\frac18\cos (4kx) -\frac12 \cos(2kx)$. Then,
\begin{align} &\int_0^\infty \frac{\sin^4(7x)-\sin^4(5x)}{x}dx\\ =&\frac18\int_0^\infty \frac{\cos(28x)-\cos(20x)}{x}dx -\frac12\int_0^\infty \frac{\cos(14x)-\cos(10x)}{x}dx\\ = &(\frac18-\frac12)\ln\frac57=\frac38\ln\frac75 \end{align}
where the result $\int_0^\infty \frac{\cos(ax)-\cos(bx )}{x}dx=\ln\frac b a$is used.
Finding $\int_0^\infty \frac{\cos(ax)-\cos(bx)}{x}dx$