Integration Calculus

[cricket21]

Hi, I would appreciate help on this question.
If

, then there is a unique real number

such that

Express

as a rational function of

and

.

Thanks

 

[soroban]

Hello, cricket21!

\(\displaystyle \text{If }-1 < x,y < 1,\,\text{ then there is a unique real number } z\text{such that:}\)

. . \(\displaystyle \displaystyle \int^x_0\frac{dt}{t^2+1} + \int^y_0\frac{dt}{t^2+1} \;=\;\int^z_0\frac{dt}{t^2+1}\)

\(\displaystyle \text{Express }z\text{ as a rational function of }x\text{ and }y.\)

Integrating, we have: .\(\displaystyle \arctan t\bigg]^x_0 + \arctan t\bigg]^y_0 \;=\;\arctan t\bigg]^z_0\)

. . . . . . . . . . . . . . . . . . . \(\displaystyle \arctan x + \arctan y \;=\;\arctan z \)


Take the tangent of both sides:

. . \(\displaystyle \tan(\arctan z) \;=\;\tan(\arctan x + \arctan y)\)

. . \(\displaystyle \tan(\arctan z) \;=\;\dfrac{\tan(\arctan x) + \tan(\arctan y)}{1 - \tan(\arctan x)\tan(\arctan y)}\)

Therefore: . \(\displaystyle z \;=\;\dfrac{x+y}{1-xy}\)

 

[cricket21]

Thank you