# Integral involving square root and exp

• nicnicman
In summary, this person is trying to solve a homework equation using u-substitution, but is stuck and needs help with partial fractions.
nicnicman

## Homework Statement

$\int$$\frac{dx}{\sqrt{e^{x} + 1}}$

## Homework Equations

Using u-substitution

## The Attempt at a Solution

Let u = $\sqrt{e^{x} + 1}$ $\Rightarrow$ u$^{2} - 1$ = e$^{x}$
Then, du = $\frac{e^{x} dx}{2\sqrt{e^{x} + 1}}$ $\Rightarrow$ dx = $\frac{2u du}{u^{2}-1}$

So, $\int$$\frac{dx}{\sqrt{e^{x} + 1}}$ = $\int$$\frac{2u du}{u(u^{2}-1)}$

But, I'm stuck at this point. I think I want to break it up into two simpler integrals, but I'm not sure how to do this. Any suggestions would be greatly appreciated!

nicnicman said:

## Homework Statement

$\int$$\frac{dx}{\sqrt{e^{x} + 1}}$

## Homework Equations

Using u-substitution

## The Attempt at a Solution

Let u = $\sqrt{e^{x} + 1}$ $\Rightarrow$ u$^{2} - 1$ = e$^{x}$
Then, du = $\frac{e^{x} dx}{2\sqrt{e^{x} + 1}}$ $\Rightarrow$ dx = $\frac{2u du}{u^{2}-1}$

So, $\int$$\frac{dx}{\sqrt{e^{x} + 1}}$ = $\int$$\frac{2u du}{u(u^{2}-1)}$

But, I'm stuck at this point. I think I want to break it up into two simpler integrals, but I'm not sure how to do this. Any suggestions would be greatly appreciated!

Partial fractions is what you want. Factor the denominator.

Okay.

So, $\int$$\frac{dx}{\sqrt{e^{x} + 1}}$ = $\int$$\frac{2u du}{u(u^{2}-1)}$ = $\int$$\frac{2du}{(u+1)(u-1)}$

And now I'm stuck again.

nicnicman said:
Okay.

So, $\int$$\frac{dx}{\sqrt{e^{x} + 1}}$ = $\int$$\frac{2u du}{u(u^{2}-1)}$ = $\int$$\frac{2du}{(u+1)(u-1)}$

And now I'm stuck again.

Partial fractions! ##\frac{1}{(u+1)(u-1)}=\frac{A}{u+1}+\frac{B}{u-1}## for some constants A and B. Find those constants.

Okay. I think I've got it.

So, $\int$$\frac{dx}{\sqrt{e^{x} + 1}}$ = $\int$$\frac{2u du}{u(u^{2}-1)}$ = $\int$$\frac{2du}{(u+1)(u-1)}$ = $\int$$\frac{du}{u-1}$ - $\int$$\frac{du}{u-1}$ = ln|u-1| - ln|u+1| = ln$\frac{|u-1|}{|u+1|}$ where u = $\sqrt{e^{x}+1}$

= ln$\frac{\sqrt{e^{x}+1}-1}{\sqrt{e^{x}+1}+1}$

Thank you for all your help.

## 1. What is an integral involving square root and exponential functions?

An integral involving square root and exponential functions is an expression that involves the integration of a square root function and an exponential function. It can be written in the form ∫ √(ax+b) e^(cx+d) dx, where a, b, c, and d are constants.

## 2. How do you solve an integral involving square root and exponential functions?

To solve an integral involving square root and exponential functions, you can use integration techniques such as substitution, integration by parts, or trigonometric substitution. It is important to choose the appropriate method based on the form of the integral.

## 3. What are the applications of integrals involving square root and exponential functions?

Integrals involving square root and exponential functions have various applications in physics, engineering, and economics. They are used to calculate quantities like work, velocity, and displacement in systems with varying forces or accelerations.

## 4. Are there any special cases of integrals involving square root and exponential functions?

Yes, there are special cases of integrals involving square root and exponential functions, such as when the exponential function is raised to a power or when the square root function is in the denominator. These cases may require different integration techniques and may result in different solutions.

## 5. Can integrals involving square root and exponential functions be evaluated using software?

Yes, there are many mathematical software programs, such as Mathematica and Wolfram Alpha, that can evaluate integrals involving square root and exponential functions. They use advanced algorithms and techniques to find the solutions to these integrals.

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