Solving Integrals using Substitution | Cosine Functions | Integral Help Needed

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In summary, substitution is commonly used when solving integrals involving cosine functions, especially when the integral is in the form of ∫f(g(x))g'(x)dx. The general process involves identifying a substitution variable, u, and rewriting the integral in terms of u before solving. It is recommended to use a variable that will simplify when taking the derivative, such as sin(x) or tan(x). Guidelines for choosing a substitution variable include canceling out or simplifying in the derivative and not creating a more complex integral. Common mistakes to avoid include not replacing the differential with the appropriate version and not properly simplifying the integral after substitution. It is also important to check that the substitution variable cancels out correctly in the derivative.
  • #1
Natasha1
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I have been asked to state which one of the following 3 integrals I can solve using integration by substitution

(i) integral cos (x^2) dx

(ii) integral x cos x (x^2) dx

(iii) integral x^2 cos (x^2) dx

I would say that it is (ii) because x is the derivative of x^2 is that the correct answer or is there anything else I should add? Thanks in advance
 
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  • #2
That would be okay enough, I guess.
 
  • #3
Well, x isn't the derivative of [itex]x^2[/itex], but I know what you meant. Maybe you could also show how basic substitution fails on the other two.
 

FAQ: Solving Integrals using Substitution | Cosine Functions | Integral Help Needed

How do I know when to use substitution to solve an integral involving cosine functions?

Substitution is commonly used when the integral involves a function within a function, such as cosine within cosine. Additionally, if the integral is in the form of ∫f(g(x))g'(x)dx, substitution can also be used.

What is the general process for solving integrals using substitution?

The general process involves identifying a substitution variable, u, that will replace part of the integral. Then, the integral is rewritten in terms of u, and the new integral is solved. Finally, the substitution variable is replaced back in the final answer.

Can I use any substitution variable for solving integrals involving cosine functions?

While any variable can technically be used, it is recommended to use a variable that will cancel out or simplify when taking the derivative. In the case of cosine functions, common substitution variables are sin(x) or tan(x).

Are there any specific rules or guidelines for choosing a substitution variable?

Yes, there are a few guidelines to keep in mind when choosing a substitution variable. The variable should cancel out or simplify in the derivative, and it should also make the integral simpler to solve. Additionally, the substitution should not create a more complex integral than the original one.

Are there any common mistakes to avoid when using substitution to solve integrals?

One common mistake is forgetting to replace the differential, dx, with the appropriate version in terms of the substitution variable. Another mistake is not properly simplifying the integral after substitution, which can lead to incorrect answers. It is also important to check that the substitution variable cancels out correctly when taking the derivative.

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