Maclaurin Series

A function $\displaystyle f(x)$ has the following properties:

$\displaystyle f'(x) = \frac{x+1}{1+(x+1)^4}$

the first term in the Maclaurin expansion of $\displaystyle f(x)$ is 1.

(a)Find the Maclaurin expansion of $\displaystyle f(x)$ up to and including the term in $\displaystyle x^2.$

(b)Use the substitution $\displaystyle u=(x+1)^2$ to find $\displaystyle \int \frac{x+1}{1+(x+1)^4}dx.$

(c)Determine an expression for $\displaystyle f(x).$

(a)Find and simplify the Maclaurin expansion, up to and including the term in $\displaystyle x^3$, for:

(i) $\displaystyle e^{2x}$

(ii) $\displaystyle \sin 3x$

(b)Hence find the Maclaurin expansion for $\displaystyle e^{2 \sin 3x}$ up to and including the term in $\displaystyle x^3.$

(a)Find and simplify the Maclaurin expansion, up to and including the term in $\displaystyle x^4$, for $\displaystyle \cos 3x.$

(b)Hence find and simplify the Maclaurin expansion, up to and including the term in $\displaystyle x^4$, for $\displaystyle \cos^2 3x.$

(a)Find and simplify the Maclaurin expansion, up to and including the term in $\displaystyle x^{3}$, for:

(i) $\displaystyle e^{3x}$

(ii) $\displaystyle \ln(1+x).$

(b)Hence find and simplify the Maclaurin expansion, up to and including the term in $\displaystyle x^{3}$, for $\displaystyle e^{3x}\ln\left(\frac{1}{1+x}\right).$