Solving Integral of 1/sqrt(1+x^2)

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In summary: I followed.In summary, the integral $I=\int\frac{1}{\sqrt{1+x^2}}$ can be solved by substituting $x = \tan\theta$ and $dx = \sec^{2}\theta{d\theta}$, which leads to the integral $I=\int{\sec\theta{d\theta}}$. This can be further simplified by substituting $(\sec\theta + \tan\theta) = u$ and using the derivative of $\sec\theta + \tan\theta$ to solve for $du$. The final solution involves using the natural logarithm function and yields $I=\ln{\vert\sqrt{1+x^2} + x\
  • #1
karush
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$I=\int\frac{1}{\sqrt{1+x^2}}$
I followed this example but it surprised me
$x = \tan\theta \ \ dx = \sec^{2}\theta{d\theta}$
$$I=\int\left(\frac{1}{\sec\theta}\right){\sec^{2}\theta{d\theta}}
=\int{\sec\theta{d\theta}}$$

Why does this need to be done?
$$I=\int{\left(\frac{\sec^2\theta + \sec\theta\tan\theta}{\sec\theta + \tan\theta}\right){d\theta}}$$

Let, $(\sec\theta + \tan\theta) = u$
$(\sec^2\theta + \sec\theta\tan\theta)d\theta = du$

$$I=\int\frac{du}{u}$$

$$I=\ln{u}+c$$

$$I=\ln{\vert\sec\theta + \tan\theta\vert}+c$$

$$I=\ln{\vert\sqrt{1+\tan^2\theta} + \tan\theta\vert}+c$$

$I=\ln{\vert\sqrt{1+x^2} + x\vert}+c$, where $c$ is a constant
 
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  • #2
karush said:
$I=\int\frac{1}{\sqrt{1+x^2}}$
I followed this example but it surprised me
$x = \tan\theta \ \ dx = \sec^{2}\theta{d\theta}$
$$I=\int\left(\frac{1}{\sec\theta}\right){\sec^{2}\theta{d\theta}}
=\int{\sec\theta{d\theta}}$$

Why does this need to be done?
$$I=\int{\left(\frac{\sec^2\theta + \sec\theta\tan\theta}{\sec\theta + \tan\theta}\right){d\theta}}$$

Let, $(\sec\theta + \tan\theta) = u$
$(\sec^2\theta + \sec\theta\tan\theta)d\theta = du$

$$I=\int\frac{du}{u}$$

$$I=\ln{u}+c$$

$$I=\ln{\vert\sec\theta + \tan\theta\vert}+c$$

$$I=\ln{\vert\sqrt{1+\tan^2\theta} + \tan\theta\vert}+c$$

$I=\ln{\vert\sqrt{1+x^2} + x\vert}+c$, where $c$ is a constant

They did it because they recognised that $\displaystyle \begin{align*} \frac{\mathrm{d}}{\mathrm{d}\theta}\,\left[ \sec{\left( \theta \right) } + \tan{ \left( \theta \right) } \right] = \sec{\left( \theta \right) } \tan{ \left( \theta \right) } + \tan^2{ \left( \theta \right) } \end{align*}$. As you say, this is not obvious. I personally would have written

$\displaystyle \begin{align*} \sec{ \left( \theta \right) } &= \frac{1}{\cos{\left( \theta \right) } } \\ &= \frac{ \cos{ \left( \theta \right) } }{\cos^2{ \left( \theta \right) } } \\ &= \frac{\cos{ \left( \theta \right) }}{1 - \sin^2{ \left( \theta \right) }} \end{align*}$

and then substitute $\displaystyle \begin{align*} u = \sin{ \left( \theta \right) } \implies \mathrm{d}u = \cos{ \left( \theta \right) } \,\mathrm{d}\theta \end{align*}$.On second thought, at the very start I would just substitute $\displaystyle \begin{align*} x =\sinh{(t)} \implies \mathrm{d}x = \cosh{(t)} \,\mathrm{d}t \end{align*}$...

$\displaystyle \begin{align*} \int{ \frac{1}{\sqrt{1 + x^2}}\,\mathrm{d}x} &= \int{ \frac{1}{\sqrt{1 + \sinh^2{(t)}}}\,\cosh{(t)}\,\mathrm{d}t } \\ &= \int{ \frac{\cosh{(t)}}{\sqrt{\cosh^2{(t)}}}\,\mathrm{d}t } \\ &= \int{ \frac{\cosh{(t)}}{\cosh{(t)}}\,\mathrm{d}t } \\ &= \int{ 1\,\mathrm{d}t } \\ &= t + C \\ &= \textrm{arsinh}\,\left( x \right) + C \end{align*}$
 
  • #3
Thanks that was very helpful..

Not sure why some of these examples
go off on weird trails

Your method makes more sense
 

FAQ: Solving Integral of 1/sqrt(1+x^2)

What does the integral of 1/sqrt(1+x^2) represent?

The integral of 1/sqrt(1+x^2) represents the area under the curve of the function 1/sqrt(1+x^2), also known as the inverse hyperbolic sine function. It is commonly used in physics and engineering to calculate the work done by a force acting on an object.

Why is it important to solve integrals?

Solving integrals is important because it allows us to find the exact values of quantities such as area, volume, and work, which are crucial in many fields of science and engineering. Integrals also help us to model and understand real-world phenomena, making them an essential tool in scientific research.

What is the general approach to solving integrals?

The general approach to solving integrals is to first identify the type of integral, such as a definite or indefinite integral, and then use mathematical techniques such as substitution, integration by parts, or partial fractions to simplify the integral. Once simplified, the integral can be solved using the fundamental theorem of calculus.

Can the integral of 1/sqrt(1+x^2) be solved by hand?

Yes, the integral of 1/sqrt(1+x^2) can be solved by hand using the substitution method. It involves substituting u = 1+x^2, which simplifies the integral and makes it easier to solve. However, for more complex integrals, it may be necessary to use numerical methods or computer software to find the solution.

Are there any real-world applications of the integral of 1/sqrt(1+x^2)?

Yes, the integral of 1/sqrt(1+x^2) has many real-world applications, such as in physics to calculate the work done by a force acting on an object, in finance to calculate the present value of a continuous stream of payments, and in engineering to calculate the deflection of a beam under a distributed load. It is also used in signal processing and image recognition algorithms.

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