I have to determine the transpose $D^tf$ of linear transformation $D\in L(V,V)$, where $V=\mathcal{P}(\mathbb{R})$, for $a, b, \in \mathbb{R}$, $f\in V^{*}$ defined by:
$$f(p)=\int_a^bp(x)dx.$$
Since the $p(x)=a_0+a_1x+a_2x^2+\ldots+a_nx^n+\ldots$ and $D(p)=a_1+2a_2x^1+\ldots+na_nx^{n-1}+\ldots$ I try to do by definition of $D^tf(p(x))=f(D(p))=\int_a^bD(p(x))dx=\int_a^b(a_1+2a_2x^1+\ldots+na_nx^{n-1}+\ldots)dx.$
My question is:
1- This interpretation already response the question, or is necessary desenvolve the calculus of integral?
2- For example, Do I need present the following development?$\int_a^b(a_1+2a_2x^1+\ldots+na_nx^{n-1}+\ldots)dx=a_1x+a_2x^2+\ldots+a_nx^{n}+\ldots|_a^b=a_1(b-a)+a_2(b^2-a^2)+\ldots+a_n(b^n-a^n)+\ldots=\displaystyle\sum_{i=1}^\infty a_i(b^i-a^i).$