Question

Politehnica admitere AM 217 integrale​

Answer 1

$\displaystyle \\ \int_{0}^1\sqrt{1+\sqrt[3]{x}}\, dx = m\int_{1}^2(x-1)^2\sqrt x\, dx \\ \\ \text{Pentru prima integrala: }\\\sqrt[3]{x} = t \Rightarrow x = t^3 \Rightarrow dx = 3t^2dt\\ x = 0 \Rightarrow t = 0 \\ x = 1 \Rightarrow t = 1\\ \\ \int_{0}^1\sqrt{(1+t)}\cdot 3t^2 \,dt = m \int_{1}^2\sqrt{1+\sqrt[3]{x}}\, dx \\ 3\int_{0}^1 t^2\sqrt{(1+t)}\,dt =m \int_{1}^2\sqrt{1+\sqrt[3]{x}}\, dx$

$\text{Tot pentru prima integrala: }\\1+t = x \Rightarrow t = x-1 \Rightarrow dt = dx \\ t = 1 \Rightarrow x = 2 \\ t = 0 \Rightarrow x = 1\\ \\ \Rightarrow \displaystyle 3\int_{1}^2 (x-1)^2\sqrt{x}\, dx = m \int_{1}^2 (x-1)^2\sqrt{x}\, dx \\ \\ \Rightarrow \boxed{m = 3}$