Integration and radicals

Solution as suggested by @Aaghaz Mahajan

Let $y = \sqrt[2017]{x\sqrt[2017]{x\sqrt[2017]{x\cdots}}}$ .

Then, we have:

$\begin{aligned} y & = \sqrt[2017]{xy} & \small \color{#3D99F6} \text{Raising both sides to the power of }2017 \\ y^{2017} & = xy \\ y^{2016} & = x & \\ \implies y & = x^{\frac 1{2016}} \end{aligned}$

Now, we have:

$\begin{aligned} I & = \int_0^1 \sqrt[2017]{x\sqrt[2017]{x\sqrt[2017]{x\cdots}}} \ dx \\ & = \int_0^1 y \ dx \\ & = \int_0^1 x^{\frac 1{2016}} dx \\ & = \left[\frac {2016}{2017}x^{\frac {2017}{2016}} \right]_0^1 \\ & = \frac {2016}{2017} \\ & = 1 - \frac 1{2017} \end{aligned}$

$\implies k = \boxed{2017}$