Solving Integration: x cos^2(x) / (1 + x)^1/2 - Step by Step Guide

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In summary: That's hardly implying "certainty" that you should use Simpson's method! But if this is in a text on numerical methods and is asking you to integrate a function, I'd be very surprised if Simpson's method was not explained just a few pages earlier.
  • #1
optics.tech
79
1
Hello everyone,

This one makes me very confuse!

Can someone please tell me how to solve the following integration?

Homework Statement


Solve the following integration:


Homework Equations


integral (x cos^2(x) / (1 + x)^1/2) dx


The Attempt at a Solution


Confuse

If above equation isn't clear enough, please see below image:

[PLAIN]http://img148.imageshack.us/img148/6692/70786744.png

Thank you very much for your help.
 
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  • #2
Are you sure that this integral is solvable?I think it requires the use of incomplete Gamma Function.But then this integral is not solvable for real x
 
  • #3
I've tried to solve it by integration techniques in the very basic english calculus textbook but I couldn't find a method to solve it.

This exercise is from Exploring Numerical Methods - An Introduction to Scientific Computing Using MATLAB book by Peter Linz and Richard L. C. Wang.

I couldn't find any hints on above book too.

Maybe I should read further chapters.
 
  • #4
A problem like this in a book on numerical methods is most likely intended to be solved by numerical methods, such as the Simpson's Method.
 
  • #5
A problem like this in a book on numerical methods is most likely intended to be solved by numerical methods, such as the Simpson's Method.

Hello Mark44,

Why can you with certainty point to The Simpson's method?

Please I want to know it too especially algebraically.

Thank you

Huygen
 
  • #6
The integral, surprisingly enough, doesn't require use of the incomplete gamma, but it does need the imaginary error function erfi(x) defined as...

[tex]erfi(x) = \frac{\sqrt{\pi} \int e^{-\left(i x\right)^2}}{2 i}[/tex]

At least I believe that's how it's defined. It's also functionally equal to...

[tex]erfi(x) = \frac{erf\left(i x\right)}{i}[/tex]

Proof by computer is found here.
 
  • #7
Simpson's rule is a method of approximating an integral by evaluating the function at various points:

http://en.wikipedia.org/wiki/Simpson's_rule

Learning about things like this is what a Numerical Methods class is basically about.
 
  • #8
Maybe its typo and it is cos 2 x inspite of cos ^2 x
 
  • #9
snshusat161 said:
Maybe its typo and it is cos 2 x inspite of cos ^2 x
I doubt it very much, as it is written as cos2(x) in the integral.
 
  • #10
Mark44 said:
I doubt it very much, as it is written as cos2(x) in the integral.

but such question is very difficult to solve in precalculus mathematics. I don't know the solution or any way to solve it cause I had left solving maths from last 3 months. (holiday's going) but after going through other members answer I'm forced to think there is some typo.

@OP, btw, in which class you are?
 
  • #11
snshusat161 said:
but such question is very difficult to solve in precalculus mathematics.
Not only that, it's pretty much impossible. Courses before calculus don't work with differentiation or integration.
snshusat161 said:
I don't know the solution or any way to solve it cause I had left solving maths from last 3 months. (holiday's going) but after going through other members answer I'm forced to think there is some typo.

@OP, btw, in which class you are?
The OP said the exercise is from a book on numerical methods.
 
  • #12
optics.tech said:
Hello Mark44,

Why can you with certainty point to The Simpson's method?
Why "certainty"? Mark44 said "most likely intended to be solved by numerical methods, such as the Simpson's Method" (emphasis mine).

That's hardly implying "certainty" that you should use Simpson's method! But if this is in a text on numerical methods and is asking you to integrate a function, I'd be very surprised if Simpson's method was not explained just a few pages earlier.

Please I want to know it too especially algebraically.

Thank you

Huygen
 
  • #13
Maybe its typo and it is cos 2 x inspite of cos ^2 x

No, I'm sure I was typed the correct equation.

Why "certainty"? Mark44 said "most likely intended to be solved by numerical methods, such as the Simpson's Method" (emphasis mine).

That's hardly implying "certainty" that you should use Simpson's method! But if this is in a text on numerical methods and is asking you to integrate a function, I'd be very surprised if Simpson's method was not explained just a few pages earlier.

Yes, I'm sure that there is no The Simpson's method is explained before.

but such question is very difficult to solve in precalculus mathematics. I don't know the solution or any way to solve it cause I had left solving maths from last 3 months. (holiday's going) but after going through other members answer I'm forced to think there is some typo.

Yes, as I was already told before, I wasn't able to solve the equation with any basic integration techniques on basic calculus textbook.
 
  • #14
I wonder if I may typed the complete exercise of this in this forums?
 
  • #15
Then I will go ahead and move this thread to the "Calculus" homework section. An integral itself is hardly "pre-calculus".
 
  • #16
I wonder if I may typed the complete exercise of this in this forums?

Then I will go ahead and move this thread to the "Calculus" homework section. An integral itself is hardly "pre-calculus".

If I'm not mistake on understanding, I deem it as yes.
 
  • #17
Here are the exercises from the book:

1. Based on the results of Table 1.1. estimate what value of n would be required to evaluate the integral (1.2) to an accuracy of eight decimal digits.

[PLAIN]http://img571.imageshack.us/img571/3623/98860439.png

2. Write a computer program that evaluates integrals by the rectangular rule (1.5). Use this program to find an approximate value of the volume for the solid of revolution given by (1.1) in Example 1.1 with r(x) = [tex]\frac{1}{1+\sqrt{x^5}}[/tex]

[PLAIN]http://img683.imageshack.us/img683/5971/94056025.png

3. Use the program from Exercise 2 to correctly compute

[PLAIN]http://img641.imageshack.us/img641/3782/30764364.png

to three decimal digits. Provide arguments that lead you to believe that your answer meets the accuracy requirement.
 
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  • #18
What's your question?
 

Related to Solving Integration: x cos^2(x) / (1 + x)^1/2 - Step by Step Guide

1. What is integration and why is it important?

Integration is a mathematical process that involves finding the antiderivative of a given function. It is important because it allows us to solve various problems in physics, engineering, and other fields that involve calculating areas, volumes, and other quantities.

2. How do you solve the integral of x cos^2(x) / (1 + x)^1/2?

To solve this integral, we can use the substitution method. Let u = 1 + x, then du = dx. The integral becomes ∫x cos^2(x) / (1 + x)^1/2 dx = ∫(u-1) cos^2(u-1) / u^1/2 du. Using the power rule, we can simplify this to ∫(u-1) cos^2(u-1) / u^1/2 du = ∫(u-1) (1+cos(2u-2)) / u^1/2 du. Using the distributive property, we get ∫(u-1) / u^1/2 du + ∫(u-1) cos(2u-2) / u^1/2 du. The first integral can be solved using the power rule, while the second integral can be solved using the u-substitution method. Finally, we can combine the results to get the final answer.

3. What are the steps involved in solving an integration problem?

The steps involved in solving an integration problem are as follows: 1. Identify the type of integral (definite or indefinite). 2. Simplify the integrand if possible. 3. Use appropriate integration techniques (substitution, integration by parts, partial fractions, etc.) to solve the integral. 4. Check the solution by differentiating it. 5. Add the constant of integration if the integral is indefinite. 6. If the integral is definite, evaluate it using the limits of integration.

4. What is the substitution method and when is it used?

The substitution method, also known as the u-substitution method, is a technique used to solve integrals that involve a function within a function. It is used when the integrand contains a variable that can be substituted by a new variable to simplify the integral. The new variable is chosen in such a way that it makes the integral easier to solve.

5. Can you provide a step-by-step guide for solving x cos^2(x) / (1 + x)^1/2?

Step 1: Identify the type of integral (indefinite or definite)
Step 2: Simplify the integrand if possible
Step 3: Use the substitution method by letting u = 1 + x
Step 4: Rewrite the integral in terms of u: ∫(u-1) cos^2(u-1) / u^1/2 du
Step 5: Expand the integral using the distributive property: ∫(u-1) (1+cos(2u-2)) / u^1/2 du
Step 6: Split the integral into two parts: ∫(u-1) / u^1/2 du + ∫(u-1) cos(2u-2) / u^1/2 du
Step 7: Solve the first integral using the power rule: 2/3 u^3/2 - 2 u^1/2 + C
Step 8: Solve the second integral using the u-substitution method: ∫cos(2u-2) du = 1/2 sin(2u-2) + C
Step 9: Combine the results: 2/3 u^3/2 - 2 u^1/2 + 1/2 sin(2u-2) + C
Step 10: Substitute back u = 1 + x: 2/3 (1 + x)^3/2 - 2 (1 + x)^1/2 + 1/2 sin(2 + 2x) + C
Step 11: Simplify the answer (if needed).

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