Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2006 / Paper 2 / 6 / a / Solution

For $x>0$, the kernel is smooth between the two positive endpoints. The Leibniz integral rule gives both endpoint terms and the parameter derivative:The remaining integral equals $[\cos (x^3)-\cos (x^4)]/x$, soThe substitution $u=xt$ gives the same result directly from limits $x^3,x^4$. Negative $x$ would make the original interval cross the nonintegrable $1/t$ singularity, so it is not part of this ordinary-integral domain.