# Find this Integration: Is There a Simplified Form for This Integral?

• utkarshakash
In summary, the conversation discusses using substitution to simplify the integral ##\int_0^1 \frac{xe^{\tan^{-1}x}}{\sqrt{1+x^2}} dx##. The attempt at a solution involves substituting ##\tan^{-1}(x)=t##, but the resulting integral becomes more complex. The conversation suggests using the substitution ##\sqrt{1+x^2}## to simplify the integral.

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## Homework Statement

$\displaystyle \int_0^1 \dfrac{xe^{tan^{-1}x}}{\sqrt{1+x^2}} dx$

## The Attempt at a Solution

Let tan^-1 (x) = t
x = tant
dt=dx/sqrt{1+x^2}
The integral then reduces to

$\displaystyle \int_0^{\pi/4} tante^tdt$

Applying integration by parts by taking tant as 1st function

$tant e^t - \displaystyle \int sec^2te^t dt$

utkarshakash said:

## Homework Statement

$\displaystyle \int_0^1 \dfrac{xe^{tan^{-1}x}}{\sqrt{1+x^2}} dx$

## The Attempt at a Solution

Let tan^-1 (x) = t
x = tant
dt=dx/sqrt{1+x^2}
The integral then reduces to

$\displaystyle \int_0^{\pi/4} tante^tdt$

Applying integration by parts by taking tant as 1st function

$tant e^t - \displaystyle \int sec^2te^t dt$

$$\frac{d}{dx}\left(\arctan(x)\right) ≠ \frac{1}{\sqrt{1+x^2}}$$
To use the substitution, multiply and divide by ##\sqrt{1+x^2}##.

## What is the purpose of finding an integration?

Finding an integration refers to the process of finding a mathematical solution that represents the area under a curve or the sum of two or more functions. This is often used in physics, engineering, and other scientific fields to calculate quantities such as displacement, velocity, and acceleration.

## What are the different methods for finding an integration?

There are several methods for finding an integration, including the fundamental theorem of calculus, integration by substitution, integration by parts, and partial fraction decomposition. Each method is useful for different types of integrals and can be used to simplify complex mathematical expressions.

## How do you know when to use which method for finding an integration?

The method used for finding an integration depends on the form of the integral and the functions involved. For example, integration by substitution is useful for integrals with nested functions, while integration by parts is useful for integrals with products of functions. It is important to understand the properties and applications of each method to determine which one is best suited for a particular integral.

## What are some common mistakes when finding an integration?

Some common mistakes when finding an integration include forgetting to add the constant of integration, missing a negative sign, or making a mistake while applying a specific method. It is important to double-check all steps and calculations to ensure an accurate solution.

## How can I improve my skills in finding integrations?

Practice is key to improving your skills in finding integrations. It is also helpful to review the properties and applications of each method, as well as common mistakes to avoid. Additionally, seeking help from a tutor or attending a workshop can also aid in improving your skills in this area.