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

[optics.tech]

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.

 

[sagardip]

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

 

[optics.tech]

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.

 

[Mark44]

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.

 

[optics.tech]

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

 

[Char. Limit]

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

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

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

erfi(x) = \frac{erf\left(i x\right)}{i}

Proof by computer is found here.

 

[hgfalling]

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.

 

[snshusat161]

Maybe its typo and it is cos 2 x inspite of cos ^2 x

 

[Mark44]

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.

 

[snshusat161]

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?

 

[Mark44]

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.

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.

 

[HallsofIvy]

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

 

[optics.tech]

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.

 

[optics.tech]

I wonder if I may typed the complete exercise of this in this forums?

 

[HallsofIvy]

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

 

[optics.tech]

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.

 

[optics.tech]

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) = \frac{1}{1+\sqrt{x^5}}

[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.

 

[Mark44]

What's your question?