Integrate: $\frac{dx}{(R^{2}+x^{2})^{3/2}}$

[hasan_researc]

Homework Statement

$$\int\frac{dx}{(R^{2}+x^{2})^{3/2}}$$

Homework Equations

The Attempt at a Solution

$$\mbox{Let }x = R\\tan\theta \Rightarrow dx = R\\sec^{2}d\theta\\ x^{2} = R^{2}\\tan^{2}\theta\\ \Rightarrow R^{2} + x^{2} = R^{2}\\( 1 + tan^{2}\theta)\\ \Rightarrow R^{2} + x^{2} = R^{2}\\sec^{2}\\ \Rightarrow (R^{2} + x^{2})^{3/2} = R^{3}\\sec^{3}\theta\\ \\ Therefore, \int\frac{dx}{(R^{2}+x^{2})^{3/2}}\\ =\int\frac{R\\sec^{2}\theta}{R^{3}\\sec^{3}\theta}\\d\theta\\ =\frac{1}{R^{2}}\\ \int^{\infty}_{-\infty}cos\theta\\ d\theta\\ =\frac{1}{R^{2}}\stackrel{lim}{n\rightarrow\infty} \int^{N}_{-N}cos\theta\\ d\theta\\ =\frac{1}{R^{2}}\stackrel{lim}{n\rightarrow\infty}(sin\theta)\mid^{N}_{-N}\\ =\frac{1}{R^{2}}\stackrel{lim}{n\rightarrow\infty}(sin N + sin N) =\frac{2}{R}\stackrel{lim}{n\rightarrow\infty}sin N$$

and then?

 

[hasan_researc]

Homework Statement

$$\int\frac{dx}{(R^{2}+x^{2})^{3/2}}$$

Homework Equations

The Attempt at a Solution

$$\mbox{Let }x = R\\tan\theta \Rightarrow dx = R\\sec^{2}d\theta$$

$$x^{2} = R^{2}\\tan^{2}\theta$$

$$\Rightarrow R^{2} + x^{2} = R^{2}\\( 1 + tan^{2}\theta)$$

$$\Rightarrow R^{2} + x^{2} = R^{2}\\sec^{2}\theta$$

$$\Rightarrow (R^{2} + x^{2})^{3/2} = R^{3}\\sec^{3}\theta$$

$$\mbox{Therefore, }\int\frac{dx}{(R^{2}+x^{2})^{3/2}}$$

$$=\int\frac{R\\sec^{2}\theta}{R^{3}\\sec^{3}\theta}\\d\theta$$

$$=\frac{1}{R^{2}}\\ \int^{\infty}_{-\infty}cos\theta\\ d\theta$$

$$=\frac{1}{R^{2}}\stackrel{lim}{n\rightarrow\infty} \int^{N}_{-N}cos\theta\\ d\theta$$

$$=\frac{1}{R^{2}}\stackrel{lim}{n\rightarrow\infty}(sin\theta)\mid^{N}_{-N}$$

$$=\frac{1}{R^{2}}\stackrel{lim}{n\rightarrow\infty}(sin N + sin N)$$

$$=\frac{2}{R^{2}}\stackrel{lim}{n\rightarrow\infty}sin N$$

and then?

 

[Char. Limit]

Why did you suddenly switch from indefinite to definite (across the entire real axis, no less) integrals partway through? I would assume that you continue to use the indefinite integral.

 

[vela]

Fixed your LaTex.

$$\begin{align*} \int \frac{dx}{(R^2+x^2)^{3/2}} & = \int\frac{R\sec^2\theta}{R^3\sec^3\theta} d\theta \\ & = \frac{1}{R^2} \int^{\infty}_{-\infty}\cos\theta\,d\theta \\ & = \frac{1}{R^2} \lim_{N \to \infty} \int^N_{-N}\cos\theta\,d\theta \\ & = \frac{1}{R^2} \lim_{N \to \infty} (\sin\theta)\mid^N_{-N} \\ & = \frac{1}{R^2} \lim_{N \to \infty} (\sin N + \sin N) \\ & = \frac{2}{R^2} \lim_{N \to \infty} \sin N \end{align*}$$

You need to change the limits of the integral when you change variables.

 

[hasan_researc]

I thought that an indefinite integral is just a definite integral with limits that are infinity.
Isn't that true?

 

[Fightfish]

I thought that an indefinite integral is just a definite integral with limits that are infinity.
Isn't that true?

Nope, that is unfortunately not true at all.

 

[hasan_researc]

But $$\theta$$ from $$\infty$$ to $$-\infty$$ is the same as $$x$$ from $$\infty$$ to $$-\infty$$. Isn't it?

 

[hasan_researc]

What is an indefinite integral, then?

 

[zcd]

$$\lim_{\theta\to\infty}{\tan(\theta)}$$ doesn't exist. You have a better chance reversing the substitution before taking the limits.

 

[Fightfish]

An indefinite integral simply produces the antiderivative of the function being integrated.
$$\int f(x) dx = F(x) + c$$
where differentiating F(x) yields the function f(x).

Meanwhile, a definite integral of a function over a specified interval is equal to the difference between the values of the antiderivative at the endpoints of the interval.
$$\int_{a}^{b}f(x) dx = F(b) - F(a)$$

One can view the antiderivative as the basis with which we employ the fundamental theorem of calculus to compute definite integrals.

 

[Mark44]

I thought that an indefinite integral is just a definite integral with limits that are infinity.
Isn't that true?

As already noted, that's not true. You might be thinking of an improper integral.

 

[Redbelly98]

Moderator's note -- please continue the discussion here:

https://www.physicsforums.com/showthread.php?t=412765