1.1: Binomial series and algebraic integration

Given that $\displaystyle (2+3x)^{-1}$ can be expressed in the form $\displaystyle p(1+qx)^{-1}$, where $\displaystyle p$ and $\displaystyle q$ are constants,

(a) find the value of $\displaystyle p$ and the value of $\displaystyle q$.

(b) Hence expand $\displaystyle (2+3x)^{-1}$ in ascending powers of $\displaystyle x$ up to and including the term in $\displaystyle x^3$, expressing each coefficient as an exact fraction in its lowest terms.

(c) Obtain a series expansion for $\displaystyle f(x)$, in ascending powers of $\displaystyle x$ up to and including the term in $\displaystyle x^3$, expressing each coefficient as an exact fraction in its lowest terms.

(d) Hence use algebraic integration to obtain an estimate, to 4 decimal places, of