Integrate f(x)=1/(9+7*(sinx)^2): Hint & Solution

In summary, the function f(x) = 1/(9+7*(sinx)^2) can be integrated using the substitution method, with the substitution u = sinx and du = cosx dx. This leads to the integral of 1/(9+7*u^2), which can be solved using the inverse tangent function and the power rule. The final solution is (1/7)*arctan(7*sinx/9) + C.
  • #1
nikolafmf
114
0

Homework Statement



This function of one variable need to be integrated: f(x)=1/(9+7*(sinx)^2) from x=0 to x=pi, without help with calculators or computer algebra software.

Homework Equations



Have no idea.

The Attempt at a Solution



Have no idea.

I don't have any idea how to start. Please help with a hint :).
 
Last edited:
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  • #2
nikolafmf said:

Homework Statement



This function of one variable need to be integrated: f(x)=1/(9+7*(sinx)^2) from x=0 to x=pi, without help with calculators or computer algebra software.


Homework Equations



Have no idea.

The Attempt at a Solution



Have no idea.

I don't have any idea how to start. Please help with a hint :).

Do you have a textbook or course notes? If so, start there.
 
  • #3
Ray Vickson said:
Do you have a textbook or course notes? If so, start there.

Nothing found in the textbook which would be useful.
 
  • #4
It is unlikely that you have been assigned a homework question which has nothing to do with the material that has been taught in your class or written in your textbook, if you cannot find a single example of a similar looking integral being solved then it probably means you need to look again.
 
  • #5
Hi there,

How would one integrate a function like this: f(x)=1/(9+7*(sinx)^2) from x=0 to x=pi?

I have no idea how to start. Any hint or help will be highly appreciated.
 
  • #6

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  • #7
Tommaso_Russo said:
You are 28, not 18, so...

What should that imply?

Anyway, I want to integrate the function by pencil and paper, not by computer algebra software, calculator or any electronic device. I will be very thankful about any hint.
 
  • #8
Take a look at a table of integrals and see if your function is similar to one of the integrals in the table.

Just eyeballing what you have, either a trigonometric or hyperbolic substitution would be used here.
 
  • #9
Divide denominator and numerator by cos^2(x). Then substitute tan^2(x) in the denominator as T (or any other variable you want). Then proceed with the procedure for solving integration by substitution.
If you have problem in solving this question, let me know.
 
  • #10
(Moderator's Note -- 2 threads merged)
 
  • #11
nikolafmf said:
Hi there,

How would one integrate a function like this: f(x)=1/(9+7*(sinx)^2) from x=0 to x=pi?

I have no idea how to start. Any hint or help will be highly appreciated.

Try writing 9+7*sin(x)^2 as 9*(sin(x)^2+cos(x)^2)+7*sin(x)^2. That might give you an idea.
 
  • #12
nikolafmf said:
What should that imply?

Anyway, I want to integrate the function by pencil and paper, not by computer algebra software, calculator or any electronic device. I will be very thankful about any hint.

Since your textbook is apparently useless, go on-line: a Google search under "integration methods" produces a host of useful websites, including http://www.mathwords.com/i/integration_methods.htm or http://tutorial.math.lamar.edu/Classes/CalcII/IntTechIntro.aspx or http://math.colorado.edu/~prestos/math2300/math2300integral.pdf . This last one seems particularly relevant to your issues.
 
Last edited by a moderator:
  • #13
You are 28, not 18, so...

nikolafmf said:
What should that imply?

I meant: you are not a young student looking for solutions to a homework; so. I tought you were a technician or a researcher needing that integral for some computations. In such cases, the fast way is just Wolfram Alpha.

nikolafmf said:
Anyway, I want to integrate the function by pencil and paper, not by computer algebra software, calculator or any electronic device. I will be very thankful about any hint.

Reading also what you have written in the merged thread, now I understand you are not looking for a solution for that particular integral, but rather for an explanation of what "integral" means...

Well, if this is the case, to solve

[itex]\int[/itex] 0 π f(x) dx

with

f(x)=1/(9+7*(sin x)2)

you have to:

1) find a primitive, AKA indefinite integral, AKA antiderivative (http://en.wikipedia.org/wiki/Antiderivative) of f(x), i.e. a function F(x) whose derivative is just f(x).

Of course, if F(x) is such a function, also F(x)+k (with any number in place of k) is a primitive of f(x), given that the derivative of a constant is zero.


2) compute F(π) - F(0). Of course, with any k, F(π) + k - [F(0) + k] will give the same numerical result.

The latter point, 2), is due to the Fundamental theorem of calculus http://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus. It's demonstration is rather intuitive, have a look.

For what concerns point 1), while finding the derivative of a function can be tedious but is straightforward using simple rules, finding a primitive is often considerably hard. There are some techniques and some tricks that can be attempted, hoping that they could lead to the solution: but knowing in advance which could be more promising is a matter of experience (acquired after several homeworks :- ). Other post in this thread have suggested you some ways.

On the contrary, once somebody or some software has found the primitive you need (Wolfram Alpha gave it in the section "indefinite integral"), checking that it is the right solutions requires only to derive it.

Zdravo
 
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Related to Integrate f(x)=1/(9+7*(sinx)^2): Hint & Solution

1. How do I integrate the function f(x)=1/(9+7*(sinx)^2)?

To integrate this function, you can use the substitution method. Let u = sin(x), then du = cos(x)dx. Rewrite the function as f(u) = 1/(9+7u^2) and use the formula ∫f(u)du = (1/√a)tan^-1(√a * u) + C. Substitute back u = sin(x) to get the final answer.

2. What is the hint for integrating f(x)=1/(9+7*(sinx)^2)?

The hint for this integration problem is to use the substitution method and to remember the formula ∫f(u)du = (1/√a)tan^-1(√a * u) + C.

3. Is there a simpler way to integrate f(x)=1/(9+7*(sinx)^2)?

Yes, you can also use the trigonometric identity cos^2(x) = 1 - sin^2(x) to rewrite the function as f(x) = 1/(9+7*(1-cos^2(x))). Then, use the substitution method with u = cos(x) to get the final answer.

4. Can I use any other method to integrate f(x)=1/(9+7*(sinx)^2)?

Yes, you can also use the double angle formula sin(2x) = 2sin(x)cos(x) to rewrite the function as f(x) = 1/(9+7*(sin(2x)/2)^2). Then, use the substitution method with u = sin(2x)/2 to get the final answer.

5. What is the solution to integrating f(x)=1/(9+7*(sinx)^2)?

The solution to this integration problem is ∫1/(9+7*(sinx)^2)dx = (1/√7)tan^-1((√7/3)*sin(x)) + C.

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