Given this density function $f(x)=(\theta+1)x^{\theta}$ in $[0,1]$
Find the estimator by the method of moments for $\theta$ and by the maximum likelihood method....
I'm trying to use parametric method of moments but I guess very wrong. What is the right procedure?
The method of moments estimator is found by taking the raw moments of the distribution and equating them with the sample moments, until a unique solution is found for the resulting system.
Here is an example using an exponential distribution: suppose $X \sim \operatorname{Exponential}(\lambda)$, where $$f_X(x) = \lambda e^{-\lambda x} \, \quad x > 0.$$ It is easy to calculate $$\operatorname{E}[X] = \int_{x=0}^\infty x f_X(x) \, dx = \int_{x=0}^\infty \lambda x e^{-\lambda x} \, dx = \frac{1}{\lambda}.$$ Now, if we are given an IID sample $$\boldsymbol X = (X_1, X_2, \ldots, X_n)$$ from this distribution, the sample mean is simply $$\bar X = \frac{1}{n} \sum_{i=1}^n X_i.$$ Therefore, the method of moments estimator for $\lambda$ arises by equating the first raw moment $\operatorname{E}[X]$ with the sample mean $\bar X$, and solving for $\lambda$: $$\tilde \lambda = \frac{n}{\sum_{i=1}^n X_i}.$$
In the case where you might want the joint method of moments estimators of a distribution with two unknown parameters, you might equate both the first raw moment $\operatorname{E}[X]$ and the second raw moment $\operatorname{E}[X^2]$ with the sample mean $\bar X$ (which is the first sample moment), and the mean sum of squares $$\overline{X^2} = \frac{1}{n} \sum_{i=1}^n X_i^2,$$ also known as the second sample moment, respectively. This is because if you have two unknown parameters, you would need to solve a system of two equations for the two unknowns, and using only the first moment will give only one equation or constraint.
In your case, you have only one parameter $\theta$, so all you need to do is compute the expectation of the distribution you are given, and equate it to the sample mean $\bar X$, and solve the resulting equation for $\theta$.