- #1

Ric-Veda

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So far, I have ∫2ydy=-∫sin(3x)/cos^2(3x)dx. Is that right? If so, how do I integrate sin(3x)/cos^2(3x)?

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- Thread starter Ric-Veda
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In summary, the conversation discusses a practice question involving an elementary differential equation with a typo. The correct form of the equation is Sin(3x)dx+2yCos^2(3x)dy=0 and the conversation focuses on finding the integral of sin(3x)/cos^2(3x). The conversation ends with a suggestion to use the Weierstrass substitution or the formula for integrating by parts to solve the integral.

- #1

Ric-Veda

- 32

- 0

So far, I have ∫2ydy=-∫sin(3x)/cos^2(3x)dx. Is that right? If so, how do I integrate sin(3x)/cos^2(3x)?

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- #2

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Is this homework? And do you mean ##\sin(3x) + 2\cos(3x) \cdot y(x) \cdot y'(x) = 0\;##?

- #3

Ric-Veda

- 32

- 0

These are practice questions for elementary differential equations. Oh...I just realized that there was a typo.fresh_42 said:Is this homework? And do you mean ##\sin(3x) + 2\cos(3x) \cdot y(x) \cdot y'(x) = 0\;##?

It's: Sin(3x)dx+2yCos^2(3x)dy=0

- #4

Ric-Veda

- 32

- 0

So far, I have ∫2ydy=-∫sin(3x)/cos^2(3x)dx. Is that right? If so, how do I integrate -∫sin(3x)/cos^2(3x)?

- #5

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E.g. it includes a paragraph for your own efforts, which shortens the process a lot. (Normally we don't want to give away ready made solutions to students but try to teach them instead.)

Your typo makes the entire equation more difficult, for otherwise it would have been simply the tangent.

You can look up many integrals here: https://de.wikibooks.org/wiki/Forme...timmte_Integrale_trigonometrischer_Funktionen

It's the wrong language but there is little beside a list of formulas. You won't need the language.

##\int \frac{\sin (3x)}{\cos^2 (3x)} = \frac{1}{3 \cos(3x)}## but I haven't checked whether it's correct.

- #6

Ric-Veda

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Thank you for the link. It translates automatically into any language so it's ok. I wanted to know the whole steps on how to solve the integral: sin(3x)/cos^2(3x) (I found the template, but don't know how to use it yet. I don't go on that much on these forums for help. It's my first time using this to ask for help)fresh_42 said:

E.g. it includes a paragraph for your own efforts, which shortens the process a lot. (Normally we don't want to give away ready made solutions to students but try to teach them instead.)

Your typo makes the entire equation more difficult, for otherwise it would have been simply the tangent.

You can look up many integrals here: https://de.wikibooks.org/wiki/Forme...timmte_Integrale_trigonometrischer_Funktionen

It's the wrong language but there is little beside a list of formulas. You won't need the language.

##\int \frac{\sin (3x)}{\cos^2 (3x)} = \frac{1}{3 \cos(3x)}## but I haven't checked whether it's correct.

- #7

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No problem. Knowing the answer one could simply differentiate. The other way might be using one of the forms ##\frac{\sin x}{\cos^2 x}=\frac{\tan x}{\cos x} ## and integrate by parts: ##\int u'v = uv - \int uv'##.Ric-Veda said:Thank you for the link. It translates automatically into any language so it's ok. I wanted to know the whole steps on how to solve the integral: sin(3x)/cos^2(3x) (I found the template, but don't know how to use it yet. I don't go on that much on these forums for help. It's my first time using this to ask for help)

And thanks for the language test. I didn't know this, but surely good to know as - you might have recognized it - there are a lot more formulas for all kind of integrals.

- #8

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Look up Weierstrass substitution.

- #9

Ric-Veda

- 32

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Thanks, I know how to do it now.fresh_42 said:No problem. Knowing the answer one could simply differentiate. The other way might be using one of the forms ##\frac{\sin x}{\cos^2 x}=\frac{\tan x}{\cos x} ## and integrate by parts: ##\int u'v = uv - \int uv'##.

And thanks for the language test. I didn't know this, but surely good to know as - you might have recognized it - there are a lot more formulas for all kind of integrals.

A differential equation is an equation that involves an unknown function and one or more of its derivatives. It is used to model relationships between quantities that are changing continuously over time or space.

Differential equations are important because they are used to describe many physical, biological, and social phenomena. They provide a powerful tool for understanding and predicting the behavior of complex systems.

There are several types of differential equations, including ordinary differential equations (ODEs), partial differential equations (PDEs), and stochastic differential equations (SDEs). ODEs involve derivatives with respect to a single independent variable, while PDEs involve derivatives with respect to multiple independent variables. SDEs also take into account random factors.

The method for solving a differential equation depends on its type and complexity. Some common techniques include separation of variables, integrating factors, and series solutions. In some cases, it may be necessary to use numerical methods or computer software to approximate a solution.

Differential equations are used in a wide range of fields, including physics, engineering, chemistry, biology, economics, and social sciences. They are used to model and analyze systems such as population growth, chemical reactions, fluid dynamics, and electrical circuits.

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