# Solve Integration Homework: Prove Integral in Attachment

• zhulihuang
In summary, the conversation is about a user seeking help with solving an integral. They have tried multiple methods such as integration by parts and trigonometric substitution, but have not been successful. They mention a substitution involving variables r and R, but it is unclear how these relate to the integral. The conversation then shifts to discussing a possible solution using arctan, but the user is unsure of how to handle a potential issue with the square root. The conversation ends with suggestions to go step-by-step through the attempted solution and use the "quote" button for easier math typing.
zhulihuang

## Homework Statement

Please prove the integral in the attachment

## The Attempt at a Solution

Tried many ways to solve it. Integration by parts, trig sub etc.
The closest one is the substitution $U=\frac{r^2}{2}-\frac{R^2/2}+x^2$, which gives some arcsin. I change it into arctan, but it is not the arctan mathematica shows. I guess it has something to do with the $\sqrt{R^2 - x^2}$, since it will blow up at R. May need delta function to handle it.

#### Attachments

• QQ??20130124222700.jpg
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Welcome to PF;
Please prove the integral in the attachment
Nice try :)
This is your homework - we can help you where you get stuck but you have to do the work.

Please give it a go and show your working as far as you get.
Try to describe how you are thinking about the problem as you go.
Don't worry about looking stupid - we've all done this before: you are among understanding, and friendly, people here.

You tried many ways- please show us at least some. If possible one where you feel you came closest to solving the problem. That will make it easier to give suggestions.

Tried many ways to solve it. Integration by parts, trig sub etc.
The closest one is the substitution ##U=\frac{r^2}{2}-\frac{R^2}{2}+x^2##, which gives some ##\arcsin##. I change it into ##\arctan##, but it is not the ##\arctan## Mathematica shows. I guess it has something to do with the ##\sqrt{R^2 - x^2}##, since it will blow up at ##R##. May need delta function to handle it.
Well your integral (off the supplied attachment) seems to be: $$\int_0^b \frac{x.dx}{\sqrt{a^2+x^2}\sqrt{b^2-x^2}}=\frac{a\arctan(b/a)}{\sqrt{a^2}}$$ ... since there is no ##r## or ##R## in the integrand, it is difficult to see your reasoning behind the substitution. From context - I'm guessing R=b and r=a?

Considering the arctan in the solution - perhaps a substitution like ##x=a\tan\theta## and a table of trig identities will be the way to go?
Did you try that?

However, HallofIvy is right - it is better for you to go step-by-step through the one you felt got the closest.
So you made the substitution: what was the next step?

(I know: it's a pain typing out the math. Use the "quote" button off the bottom of this post and see how I was able to type the math out ... you can copy and paste then make minor adjustments.)

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## What is integration and why is it important?

Integration is a mathematical process used to find the area under a curve. It is important because it is used to solve many real-world problems, such as calculating volumes, determining velocities, and finding probabilities.

## What is the process for solving an integration problem?

The process for solving an integration problem involves finding the antiderivative of the given function, evaluating it at the upper and lower limits of integration, and then subtracting the values to find the final answer.

## What is the difference between definite and indefinite integration?

Definite integration involves finding the area under a curve between specific limits, while indefinite integration involves finding a general antiderivative of a given function without any specific limits.

## What are some common integration techniques?

Some common integration techniques include u-substitution, integration by parts, trigonometric substitution, and partial fractions. These techniques involve manipulating the given function to make it easier to integrate.

## How can I check if my integration solution is correct?

You can check your integration solution by taking the derivative of the antiderivative you found. If the resulting function is the same as the original function, then your solution is correct.

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