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

[utkarshakash]

Homework Statement

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

Homework Equations

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

This has made the problem more complicated instead of simplifying.

 

[Saitama]

Homework Statement

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

Homework Equations

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

This has made the problem more complicated instead of simplifying.

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