For $k>0$, $$ \int_1^k\frac{\sin^2 x}{x^2}dx=? $$
$$ \int _1^k \frac{\sin x}{x}dx=? $$
In fact ,what I really want to know is that whether $L^{p+1}(R)$ is subset of $L^p(R)$.The $L^p(R)$ is the function space on $R$, st $$ \forall f\in L^p ,~~~|\int_R f^p dx| <+\infty $$
$L^p(\mathbb{R})$ is the set of (equivalence-classes-of with respect to being equal almost everywhere) functions $f$ such that $\int_{\mathbb{R}}|f|^p\,dx < \infty$, and not what you wrote. Moverover, given $p\geq 1$, the function $1/x^{1/p}$ when $x\geq 1$ and zero otherwise belongs to $L^q(\mathbb{R})$ for all $q>p$ but does not belong to $L^p(\mathbb{R})$. So the answer to your question is negative.