How do I know what to substitute(u-substitution)

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In summary, the conversation discusses the issue of determining what to substitute when integrating inverse trig functions. The speaker shares their experience with solving problems and how the book's suggested substitutions helped them find the correct answer. The conversation ends with a question for practice on integrating a similar function.
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MMM
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Homework Statement


Hello, I'm having trouble determining what I make u equal with most integrals that revolve around the inverse trig functions.

I knew what to let u equal in this problem ##x/\sqrt{1-x^4}## I let it equal ##x^2## and correctly solved the problem.

The problems I'm having trouble with, are problems like ##1/(a^2 + x^2)## the book said to substitute u = x/a and when I did that I found the correct answer. Sadly I had no idea that is what I was supposed to substitute.

Here is another problem I had no idea what to substitute ##1/(a^2 + (b^2)(x^2))## the book said to substitute u = bx/a and when I did that I got the correct answer. I just need some guidance to understand the kinda substitutions I need to make regarding integration involving inverse trig functions. Any help is greatly needed and highly appreciated, thanks.


Homework Equations





The Attempt at a Solution


I solved the stated problems with the book's help on what to substitute. I just had no idea on to what to substitute.
 
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  • #2
You are looking for something you know how to integrate. In this case when you see something like ## 1/(a^2 + b^2x^2) ##, your first instinct should be that it looks kind of like ## 1/(1 + x^2)## and so we want to substitute to put it on that form.

Here we note that
\begin{equation*}
\frac{1}{a^2 + b^2 x^2} = \frac{1}{a^2(1 + \frac{b^2}{a^2}x^2)}= \frac{1}{a^2} \frac{1}{1 + (\frac{b}{a}x)^2}
\end{equation*}
and the substitution suggests itself.

You see these things easier and faster as you get more and more experience.
 
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  • #3
Thanks for the help. I understand it now!
 
  • #4
To practice and since you knew how to integrate ##x/\sqrt{1 - x^4}##, how would you tackle
\begin{equation*}
\frac{ax}{\sqrt{b - cx^4}}?
\end{equation*}
 

Related to How do I know what to substitute(u-substitution)

1. How do I know when to use u-substitution?

The general rule for using u-substitution is when you have an integral that involves a function and its derivative. Another indication is when you have a function raised to a power, such as x^2 or sin(x), and no other terms in the integral.

2. What is the purpose of u-substitution in integration?

U-substitution is a technique used to simplify integrals by substituting a variable, typically denoted as u, for a more complicated expression. This makes it easier to solve the integral using known integration rules.

3. How do I choose the substitution variable u?

The substitution variable u should be chosen in a way that simplifies the integral. This can be done by identifying a function and its derivative in the integral and assigning u to the function. Other times, it may require manipulation or factoring of the integral to determine the appropriate substitution.

4. Can I use u-substitution for all integrals?

No, u-substitution is not applicable to all integrals. It is most useful when dealing with integrals involving a function and its derivative. For other types of integrals, other techniques such as integration by parts or trigonometric substitution may be more appropriate.

5. Are there any common mistakes to avoid when using u-substitution?

One common mistake is forgetting to substitute the derivative of u back into the integral after solving for u. Another mistake is incorrectly choosing the substitution variable, which can lead to a more complicated integral. It is important to carefully evaluate the integral before and after substitution to ensure the correct solution.

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