- #1
maniaku
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∫ (cos (1/x) / x^2 dx (Please show steps.)
Integrate ∫ cos(1/x)/x^2 dx: Steps & Solutions
- Thread starter maniaku
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In summary, the function being integrated is cos(1/x)/x^2. The steps to integrate this function are to use the substitution u = 1/x, find du/dx and substitute into the original function, integrate using the power rule, and then substitute back u = 1/x and simplify. The substitution method involves choosing a new variable, u, and rewriting the function in terms of u to make it easier to integrate. An example of integrating this function using the substitution method is -sin(1/x) + C. Other methods such as integration by parts or using trigonometric identities can also be used, but the substitution method is the most efficient for this function.
- #2
berkeman
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You show us *your* steps, and then we can offer tutorial help. We do not do your work for you here on the PF.
- #3
roam
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Welcome to the former People's Empire! 
[tex]\int \frac{cos \frac{1}{x}}{x^2} dx[/tex]
Why couldn't you try the method of substitution? Find an appropriate substitution, and then you need to find something that can serve as du or rather, something that can simplify the denominator.
I hope you enjoy yourself!
[tex]\int \frac{cos \frac{1}{x}}{x^2} dx[/tex]
Why couldn't you try the method of substitution? Find an appropriate substitution, and then you need to find something that can serve as du or rather, something that can simplify the denominator.
I hope you enjoy yourself!
1. What is the function being integrated?
The function being integrated is cos(1/x)/x^2.
2. What are the steps to integrate this function?
The steps to integrate this function are as follows:
- Use the substitution u = 1/x.
- Find du/dx and substitute into the original function.
- Integrate the resulting function using the power rule.
- Substitute back u = 1/x and simplify the function.
3. How do I use the substitution method to integrate this function?
The substitution method involves choosing a new variable, u, and rewriting the function in terms of u. In this case, we choose u = 1/x. Then, we find du/dx and substitute it into the function. This allows us to rewrite the function in terms of u instead of x, making it easier to integrate.
4. Can you show an example of integrating this function using the substitution method?
Sure, here's an example:
Integrate ∫ cos(1/x)/x^2 dx using the substitution method.
Solution:
Let u = 1/x, then du/dx = -1/x^2. Substituting into the original function, we get:
∫ cos(u)/x^2 * (-1/x^2) dx
= ∫ -cos(u) du
= -sin(u) + C, where C is the constant of integration.
Substituting back u = 1/x, we get the final answer:
-sin(1/x) + C
5. Are there any other methods to integrate this function?
Yes, there are other methods such as integration by parts or using trigonometric identities. However, the substitution method is the most straightforward and efficient method for this particular function.
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