Integral of cos(t)/(1+9sin^2(t)) Explained

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In summary, the integral of cos(t)/(1+9sin^2(t)) can be solved using the substitution method, where the variable u = 3sin(t) is substituted in and the inverse tangent function is used to get the final answer of (1/3)arctan(3sin(t)) + C. The substitution method is used because it simplifies the problem and the constant C represents the constant of integration. This integral can also be solved using the trigonometric identity cos^2(x) + sin^2(x) = 1, but the substitution method is preferred. It has applications in various fields, such as physics and electrical engineering, for solving problems involving periodic functions.
  • #1
americanforest
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Hey guys,

By what rules is the integral of

[tex]\frac{cos(t)}{1+9sin^2(t)}=\frac{1}{3}tan^{-1}(sin(3t))[/tex]

I know this is right, but I have no idea why and can't find any trig identities to help. Thanks.
 
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  • #2
Let [tex] u = 3\sin t [/tex]

and use the fact that [tex]\int \frac{dx}{x^{2}+a^{2}} = \frac{1}{a}\tan^{-1}\left(\frac{x}{a}\right) + C [/tex]
 
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  • #3
gotcha, thanks
 

1. What is the integral of cos(t)/(1+9sin^2(t))?

The integral of cos(t)/(1+9sin^2(t)) is a trigonometric integral that can be solved using the substitution method. By letting u = 3sin(t), the integral can be transformed into a simpler form of 1/(1+u^2). This can then be solved using the inverse tangent function, arctan(u), to get the final answer of (1/3)arctan(3sin(t)) + C.

2. Why is the substitution method used to solve this integral?

The substitution method is used to solve this integral because it simplifies the problem and makes it easier to solve. By substituting in a new variable, the integral can be transformed into a simpler form that can be solved using known integration techniques.

3. What is the significance of the constant C in the final answer?

The constant C represents the constant of integration and is added to the final answer because indefinite integrals have an infinite number of solutions. The constant C allows for all possible solutions to be included in the final answer.

4. Can this integral be solved using any other integration techniques?

Yes, this integral can also be solved using the trigonometric identity cos^2(x) + sin^2(x) = 1, which can be manipulated to get the final answer of (1/3)(tan(t) - 3tan(t)) + C. However, the substitution method is often preferred as it is more straightforward and does not require memorizing trigonometric identities.

5. How can this integral be applied in real-life situations?

This integral can be applied in various fields of science and engineering, such as physics and electrical engineering, to solve problems involving periodic functions. For example, it can be used to calculate the average power in an AC circuit or to determine the displacement of a pendulum over a certain time period.

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