How Do I Use Substitutions in Integration by Trigonometric Substitution?

  • Thread starter flyingpig
  • Start date
In summary, the conversation is about the spacing being incorrect when fonts are bigger in a LaTeX document and the attempt to solve it using trigonometric substitutions. The strategies for solving similar cases involving square roots are also discussed. The use of the fullpage package is also recommended.
  • #1
flyingpig
2,579
1

Homework Statement



Here is the problem, I don't know what happened but the spacing is incorrect when the fonts are bigger.


The Attempt at a Solution



Here is the code

\documentclass{report}

\begin{document}
\begin{center}\textbf{Integration by Trigonometric Substitution}\end{center}
\Large Consider the following integral $\int \sqrt{a^2 - x^2}\; dx$ \nolinebreak where $a > 0$ . If the integral were $\int x \sqrt{a^2 - x^2} \;dx$, the substitution $u = a^2 - x^2$ would be effective. However, as it stands, $\int \sqrt{a^2 - x^2} \;dx$ is more difficult. If we make a change of variables where we use the substitution $x = a\sin\theta$, then the identity $\sin^2\theta + \cos^2\theta = 1$ would allow us to get rid of the root sign.\\\\
Observe\\\\
$\sqrt{a^2 - x^2} = \sqrt{a^2 - a^2 \sin^2 \theta}$ $= \sqrt{a^2(1 - \sin^2\theta)} =$ $\sqrt{a^2\cos^2\theta} = a|\cos\theta|$\\\\
Notice how the substitution $u = a^2 - x^2$ would not have gotten you anywhere. So in general, we can make a substitution of the form $x = h(u)$ by using the reverse chain rule (remember this from Calculus I?!). To simplify our calculations, we assume \textit{h} has an inverse function (and hence it must not be even and is one-to-one).\\\\
So we have\\\\
$\int \textit{f}(x)\;dx$ = $\int \textit{f}(\textit{g}(u))\textit{g}'(u)\;du$\\\\
It should noted that when we made the substitution $x = a\sin\theta$, we restricted the values of $\theta$ in $\left [ \frac{-\pi}{2}, \frac{\pi}{2}\right ]$
\newpage {\noindent}%
Here are some general strategies for tackling on similar cases\\\\
Case \# 1 $\sqrt{a^2 - x^2}$\\

Use $x = a\sin\theta$ or $x = a\cos\theta$ and manipulate the identity $\sin^2\theta + \cos^2\theta =1$\\\\
Case \# 2 $\sqrt{a^2 + x^2}$\\

Use $x = a\tan\theta$ and manipulate the identity $\tan^2\theta + 1 = \sec^2\theta$\\\\
Case \#3 $\sqrt{x^2 - a^2}$\\

Use $x = a\sec\theta$ and manipulate the identity $\tan^2\theta + 1 = \sec^2\theta$\\\\
Notice that even though there are two different substitutions in the first case, either one will get you to the correct answer. This then brings us to a conclusion.\\\\

\begin{center}\textbf{There is no such thing as unique substitutions}\end{center}

\end{document}

The document is attached
 

Attachments

  • Sample Notes.pdf
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  • #2
Uh, what exactly is your problem? The spacing looks fine to me. (And could you please add some newlines in your LaTeX? It's pushing PF onto *both* my 1920 px monitors!)

By the way, use the fullpage package to use the entire page.
 
Last edited:

FAQ: How Do I Use Substitutions in Integration by Trigonometric Substitution?

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