Thread: Integral with parameter and Beta/Gamma Function

1. Integral with parameter and Beta/Gamma Function

Hello, guys ...

1. Calculate the integral $\displaystyle \int_{0}^{1}\left(\dfrac{x}{1 + \sqrt{1-x}}\right)^{\frac{3}{2}}dx$ with the help of $\displaystyle \Gamma$ and $\displaystyle B$.

2. We are given integral with a parameter $\displaystyle F(t) = \int_{1}^{2}\arctan\frac{t}{x}dx$. Find its derivative $\displaystyle F'(t)$.

$\displaystyle F'(t)=\int_1^2 \frac{\partial }{\partial t}\left(\arctan \left(\frac{t}{x}\right)\right) \, dx=\int_1^2 \frac{x}{t^2+x^2} \, dx$
$\displaystyle \int_{1}^{2}\dfrac{x}{t^2 + x^2}dx = \int_{t^2 + 1}^{t^2 + 4}\dfrac{du}{2u} = \dfrac{1}{2}\ln\left|u\right| \big|_{t^2+1}^{t^2 + 4} = \dfrac{1}{2}\ln(\dfrac{t^2+4}{t^2+1})$ Right?