Integral concerning $\sin^2$

Question

Consider the following integral :

$$I(x;a)= \int \frac{\sin^2(x+1)}{(x+1)^a}\,dx$$

Now let the $x$ runs from $0$ to $\infty$

i.e. let

$$I_p(a)= \int_0^\infty \frac{\sin^2(x+1)}{(x+1)^a}\,dx$$

Now we can see that $I_p(a)$ converges for all $\text{R}(a)>1$.

Also, for $a=1$ , $I_p(a)$ diverges .

Question 1 : If possible , can $I_p(a)$ be represented in following way :

$$I_p(a) = f(a) + \frac{C}{(a-1)}$$

Here , $f(a)$ is convergent for all $a\geqslant1$ and $C>0$

The factor $\frac{C}{(a-1)}$ explains the divergence of $I_p(a)$ at $a=1$

Question 2 ( important ):

Analogous to Question 1 : Only replace $\sin^2(x+1)$ by $\sin^2(g(x))$ in $I_p(a)$

where, $g(x)$ is an increasing function in $[1,\infty)$

Answer 1

For the first part, use the double angle formula to have $$\sin^2(x+1)=\frac 12 \left(1-\cos(2(x+1))\right)$$ Then use $\cos(t)=\frac {e^{it}+e^{-it}}2$ to face the exponential integral function.

You should end with something like $$I(a)=\int_0^\infty \frac{ \sin ^2(x+1)} {(x+1)^{a}}\,dx=\frac{1}{2 (a-1)}-\frac{1}{4} (E_a(2 i)+E_a(-2 i))$$ provided $\Re(a)>1$.