Integration of this trigonometry function

In summary, the conversation discusses a problem involving integration using the cosine function and its derivative. The participants have tried using different substitutions but have not been successful in finding a closed solution. One participant suggests using the Weierstraß substitution but another points out that the presence of both a polynomial and trigonometric function makes it impossible to use this method. The conversation ends with a question about whether the problem was written correctly and a suggestion to try replacing x with x^2 in the cosine function.
  • #1
songoku
2,340
340
Homework Statement
Find
$$\int x \sqrt{1+4 \cos^2 (x)} dx$$
Relevant Equations
High School Integration:
integration by substitution
integration by part
integration of trigonometry function
Is it possible to do the integration? That is the full question

I don't know where to start, try to use ##u=\cos x## and also ##\cos^2 (x) = \frac{1}{2} + \frac{1}{2} \cos (2x)## but failed.

Thanks
 
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  • #2
WolframAlpha and I assume that there is no closed solution. The fact that ##x## occurs as polynomial and as trigonometric function (whose derivative is not the polynomial in ##x##) makes it impossible to use things like e.g. the Weierstraß substitution.
 
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  • #3
Thank you very much fresh_42
 
  • #4
Are you 100% sure you wrote the problem down right? If instead of x in the cosine you have an x2 you could make more progress. (How much progress would depend on what the new integral is)
 
  • #5
Vanadium 50 said:
Are you 100% sure you wrote the problem down right? If instead of x in the cosine you have an x2 you could make more progress. (How much progress would depend on what the new integral is)
Yes 100% sure. I have re-checked several times when doing the question and before posting it here
 

FAQ: Integration of this trigonometry function

1. What is the definition of integration?

Integration is a mathematical process that involves finding the area under a curve on a graph. It is the reverse operation of differentiation, and it is used to calculate the total change in a quantity over a given interval.

2. What is a trigonometry function?

A trigonometry function is a mathematical function that involves the ratios of the sides of a right triangle. The most common trigonometry functions are sine, cosine, and tangent, which are used to calculate the relationship between the sides and angles of a triangle.

3. How do you integrate a trigonometry function?

To integrate a trigonometry function, you can use the trigonometric identities and substitution techniques. You can also use integration by parts or partial fractions for more complex trigonometry functions.

4. What are the applications of integrating trigonometry functions?

Integrating trigonometry functions is used in various fields of science and engineering, such as physics, astronomy, and engineering. It is also used in real-life applications, such as calculating the area under a curve in business and economics.

5. What are some common mistakes when integrating trigonometry functions?

Some common mistakes when integrating trigonometry functions include forgetting to use the chain rule, using the wrong substitution, and making mistakes in algebraic manipulations. It is important to carefully follow the steps and check your work to avoid these errors.

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