U-substitution integration help

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In summary, the conversation was about finding the indefinite integral of cos(1/2)x. The correct answer is 2sin(x/2) + c. One person was confused and tried to solve it using u-substitution and the power rule, but made a mistake in their substitution. Another person explained the correct substitution and solution. The original person then understood and thanked them.
  • #1
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Homework Statement



[tex]\int cos(1/2)x = 2sin(x/2) + c[/tex] ==> Correct Answer

This is an answer to one of my homework problems, but I don't understand why the indefinite integtral of cos(1/2)x is that. I've learned u-substitution and the power rule, but using those methods, I got: 2. The attempt at a solution[tex] u = .5x, du = .5dx \int cos(u)du = sin(u)+c = sin(.5x)+c[/tex] (wrong answer?)
 
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  • #2
You made a mistake while substituting. [tex]du = \frac{1}{2}dx[/tex] implies [tex]dx = 2 du[/tex].
 
  • #3
[tex]\int cos(1/2)x = 2sin(x/2) + c[/tex]

Yeah I'm confused. The integrand is equal to

2/sqrt(2)*x so the integral is equal to (1/sqrt(2))x^2 + C
 
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  • #4
Let [tex] u = \frac{1}{2}x [/tex]. Then [tex] du = \frac{1}{2} dx [/tex].So you have: [tex]2 \int \cos u \; du [/tex]
 
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  • #5
nealh149 said:
[tex]\int cos(1/2)x = 2sin(x/2) + c[/tex]

Yeah I'm confused. The integrand is equal to

2/sqrt(2)*x so the (1/sqrt(2))x^2 + C

What is there to be confused about? And where did you get these 'sqrt-s' from? As said before, you only have to substitute dx = 2 du and u = 1/2 x into the original integral, in order to obtain [tex]\int 2 \cos u du[/tex], which I assume you'll know how to handle.
 
  • #6
I think I got the right answer:

[tex]\int cos(1/2)x dx = 2 \int cos(.5x)(.5) + c = 2sin(.5x) + c[/tex]

nealh149 said:
[tex]\int cos(1/2)x = 2sin(x/2) + c[/tex]

Yeah I'm confused. The integrand is equal to

2/sqrt(2)*x so the integral is equal to (1/sqrt(2))x^2 + C

But I don't understand how you got that.
 
  • #7
OK, again: [tex]\int \cos(\frac{1}{2}x)[/tex] = after substituting = [tex]\int 2 \cos u du = 2 \sin u + C = 2 \sin (\frac{1}{2}x) + C[/tex] .
 
  • #8
Thank you , radou. I understand it now.

Nevermind what I said in post #6; I just realize that nealh149 was talking about another problem.
 

What is u-substitution in integration?

U-substitution, also known as the method of substitution or change of variables, is a technique used in integration to simplify the integrand and make it easier to evaluate. It involves substituting a new variable, u, for an expression within the integrand.

Why is u-substitution important in integration?

U-substitution is important because it allows us to solve a wider range of integrals, particularly those that involve composite functions or nested functions. It also helps us to evaluate integrals that cannot be solved using other techniques such as integration by parts or trigonometric substitution.

How do I know when to use u-substitution?

U-substitution is typically used when the integrand contains a function and its derivative, or when the integrand can be rewritten in the form of a composite or nested function. It can also be used to simplify integrands that contain radicals or exponential functions.

What are the steps for using u-substitution in integration?

The general steps for using u-substitution in integration are as follows:
1. Identify the expression within the integrand that is the derivative of the new variable u.
2. Substitute u for this expression, and also replace all other occurrences of the same expression with u.
3. Rewrite the remaining parts of the integrand in terms of u.
4. Evaluate the new integral using basic integration techniques.
5. Substitute back in the original variable to obtain the final solution.

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

Yes, some common mistakes to avoid when using u-substitution include:
- Forgetting to replace all occurrences of the substituted expression with u.
- Choosing the wrong expression to substitute for u.
- Not simplifying the integrand after substitution.
- Forgetting to substitute back in the original variable at the end.
It is important to carefully apply the steps of u-substitution and check your work for any potential errors.

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