Math Q&A environment: How to avoid cluttered equations?

Question

I want to write math questions and answers in the form of a book, which will be read by students. Can someone help me to create question, solution and tips environment, so that I could add the equations every time? Also I don't understand why my equations are cluttered.

I have attached an image about how the solution should look

\documentclass[12pt]{article}
%i have used few packages
\usepackage[margin= 1cm]{geometry}
\usepackage{amsmath}
\usepackage{tcolorbox}
\usepackage{multicol}
\usepackage{xcolor}
%document begins here
\begin{document}
%section heading 
\section{Questions on Basic integration}
%question with bold and box 
\begin{tcolorbox}
subsection begins here, the word question should be in bold
\subsection*{Question 1} Solve the following integral. $$\int \sec x(\sec x+\tan x) d x $$ 
\end{tcolorbox}
%solution word in bold
\subsubsection*{Solution}
this is a quick tip, to be highlighted
\textcolor{blue}{$$
\begin{array}{l}
\int \sec ^{2} x d x=\tan x \
\int \sec x \tan x=\sec x
\end{array}
$$}
%the solution

$$
\begin{aligned}
\int \sec x(&\sec x+\tan x) d x \
&=\int\left(\sec ^{2} x+\sec x \tan x\right) d x \
&=\int \sec ^{2} x d x+\int \sec x \tan x d x \
&=\tan x+\sec x+C
\end{aligned}
$$
%second question
\begin{tcolorbox}
%question 2 begins here 
\subsection*{Question 2} Solve the following integral. $$\int \frac{1}{x+x\log x} dx$$ 
\end{tcolorbox}
%solution in bold
\subsubsection*{Solution}
\textcolor{blue}{$$
\begin{array}{l}
\text { Put } 1+x \log x=t \
\text { then, } \frac{1}{x}; d x=d t
\end{array}
 $$}
 % solution begins
\begin{center}$\displaystyle{
\begin{array}{l}
\int \frac{1}{x+x \log x} d x \
\text { Simce, } \frac{1}{x} d x=d t \text { and }(1+\log x)=t \
=\int \frac{d t}{t} \
=\log (t)+C \
=\int \log (1+\log x)+C
\end{array}
}$ \end{center}
\end{document}

Answer 1

Something like this? Observe that while I provide code for a macro called \tip, I don't define separate question and answer environments, as I think that doing so wouldn't help with the clutter issue. (Actually, these environments might worsen the clutter issue.) I can only recommend that you become more familiar with the multi-line math environments of the amsmath package.

\documentclass[12pt]{article}
\usepackage[margin=2.5cm,a4paper]{geometry}
\usepackage{amsmath,tcolorbox}
\newcommand\tip[1]{\smash{\boxed{\color{blue}\array{r} #1 \endarray}}}
\begin{document}
\section{Questions on basic integration}
\begin{tcolorbox}
\subsection*{Question 1} Solve the following integral.
\quad$\displaystyle \int \sec x(\sec x+\tan x), dx $
\end{tcolorbox}
\paragraph{Solution}

\begin{align}
&\int \sec x(\sec x+\tan x), dx 
   \hspace{3cm} \tip{ \int \sec ^{2} x, dx=\tan x \[1ex]
                      \int \sec x \tan x ,dx=\sec x} \
&\quad=\int\bigl(\sec ^{2} x+\sec x \tan x\bigr), dx \
&\quad=\int \sec ^{2} x, dx+\int \sec x \tan x, dx \
&\quad=\tan x+\sec x+C
\end{align}
\begin{tcolorbox}
\subsection*{Question 2} Solve the following integral. 
\quad$\displaystyle\int \frac{1}{x+x\log x},dx$
\end{tcolorbox}
\paragraph{Solution}
\begin{align}
&\int \frac{1}{x+x \log x}, dx \
&\quad= \int \frac{1}{x(1+\log x)}, dx 
  \hspace{3cm}\tip{\text{\textbf{Tip:} Put } 1+\log x = t \[1ex]
                   \text{then } \frac{1}{x}, dx = dt} \
&\quad=\int \frac{dt}{t} \
&\quad=\log (t)+C \
&\quad=\int \log (1+\log x)+C
\end{align}
\end{document}

---

Addendum to address one of the OP's follow-up questions. One could define a \question macro as follows (in the preamble):

\newcommand\question[2]{%
   \tcolorbox 
   \subsection*{Question #1} 
   Solve the following integral.\quad$\displaystyle #2$ 
   \endtcolorbox}

and then write \question{1}{\int \sec x(\sec x+\tan x)\, dx} in the body of the document.

Answer 2

Is this what you want to have?

\documentclass[12pt]{article}
\usepackage{amsmath}
\usepackage[dvipsnames]{xcolor}
\usepackage{mdframed}
   \mdfsetup{
      leftmargin=-5,
      rightmargin=-5,
      innerleftmargin=5,% Distance from content to frame.  To align the content with
      innerrightmargin=5,% the surrouding text, the sum of leftmargin and innerleftmargin must be zero.
      linecolor=white,% The frame consists of inner-, middle-, and outerline.  linecolor changes all three.
      innertopmargin=0\topskip,
   }
   \mdtheorem[backgroundcolor=PineGreen!15!white]{tip}{Tip}%[section] % Uncomment if you want to include section number.
   \mdtheorem{quest}{Question}%[section] % Uncomment if you want to include section number.
   \mdtheorem{sol}{Solution}%[section] % Uncomment if you want to include section number.
\newcommand{\dd}[0]{\,\mathrm{d}} % puts a thin space and an upright 'd'.
\begin{document}
\section{Questions on Basic integration}
\begin{quest}
      Solve the following integral:
      \begin{equation}
         \int \sec x(\sec x+\tan x) \dd x.
      \end{equation}
   \end{quest}%
   %
   \begin{sol}
      \begin{tip}
         \begin{align}
            \int \sec^{2} x \dd x    &= \tan x + C\
            \int \sec x \tan x \dd x &= \sec x + C
         \end{align}
      \end{tip}%
      %
      \begin{align}
         \int \frac{1}{x+x \log x} \dd x  &= \int\frac{1}{x(1+\log(x))} \dd x\
         &= \int \frac{\dd t}{t} \
         &= \log (t)+C \
         &= \log (1+\log x)+C
      \end{align}
   \end{sol}
\end{document}

---

EDIT: As you asked in a comment to have no linebreak after 'Question 1', here is an updated version.

\documentclass[12pt]{article}
\usepackage{amsmath,amsthm}
\usepackage[dvipsnames]{xcolor}
\usepackage{mdframed}
\mdfsetup{
   leftmargin=-5,
   rightmargin=-5,
   innerleftmargin=5,% Distance from content to frame.  To align the content with
   innerrightmargin=5,% the surrouding text, the sum of leftmargin and innerleftmargin must be zero.
   linecolor=white,% The frame consists of inner-, middle-, and outerline.  linecolor changes all three.
   innertopmargin=0\topskip,
}
\newtheorem{quest}{Question}   % for numbered questions
\newtheorem{quest}{Question} % for unnumbered questions
\newtheorem{tip}{Tip}          % for numbered tips
\newtheorem{tip}{Tip}        % for unnumbered tips
\surroundwithmdframed[backgroundcolor=PineGreen!15!white]{tip}  % get the shiny box around (numbered) tips.
\surroundwithmdframed[backgroundcolor=PineGreen!15!white]{tip} % get the shiny box around (unnumbered) tips.
\newtheorem{sol}{Solution}     % for numbered solutions
\newtheorem{sol*}{Solution}   % for unnumbered solutions
\newcommand{\dd}[0]{,\mathrm{d}} % puts a thin space and an upright 'd'.
\begin{document}
   \section{Questions on Basic integration}
   \begin{quest}
      Solve the following integral:
      \begin{equation}
         \int \sec x(\sec x+\tan x) \dd x.
      \end{equation}
   \end{quest}%
   %
   \begin{sol}
      ~ % For some reason the tip-environment covers the word 'solution' when there is nothing between them.
      \begin{tip} Here can be some text.
         \begin{align}
            \int \sec^{2} x \dd x    &= \tan x + C\
            \int \sec x \tan x \dd x &= \sec x + C
         \end{align}
      \end{tip}%
      %
      \begin{align}
         \int \frac{1}{x+x \log x} \dd x  &= \int\frac{1}{x(1+\log(x))} \dd x\
         &= \int \frac{\dd t}{t} \
         &= \log (t)+C \
         &= \log (1+\log x)+C
      \end{align}
   \end{sol}
\end{document}