## Problem 1.

Calculate $\int_{0}^{1}\left(e^{\sqrt{x}}+e^{\sqrt[3]{x}}+e^{\sqrt[4]{x}}\right) \mathrm{d} x$.
## Solution and Grading

Let $I_{n}=\int_{0}^{1} e^{\sqrt[n]{x}} \mathrm{~d} x, n \in \mathbb{N}^{*}$ (we make the convention $\sqrt[1]{x}=x$). With the change of variable $t=e^{\sqrt[n]{x}}$, we obtain that $I_{n}=\int_{1}^{e} n \ln ^{n-1} t \mathrm{~d} t$. (2p)

Integrating by parts, we deduce that $I_{n}=n\left(\left.t \ln ^{n-1} t\right|_{1} ^{e}-\int_{1}^{e} t(n-1) \ln ^{n-2} t \cdot \frac{1}{t} \mathrm{~d} t\right)=n\left(e-I_{n-1}\right), \forall n \geq 2$. Moreover, $I_{1}=\int_{0}^{1} e^{x} \mathrm{~d} x=e-1$. (3p)

Using the found relation, $I_{2}+I_{3}+I_{4}=4 e+I_{2}-3 I_{3}=-5 e+10 I_{2}=15 e-20 I_{1}=5(4-e) \cdot(2 p)$