The Newton-Raphson method is an iterative method to find the roots of a continuous and differentiable function f(x). The idea is to start with an initial guess, x0, and then use the formula:

x1 = x0 - f(x0) / f'(x0)

where f'(x) is the derivative of the function f(x). The process is then repeated with x1 as the new guess until the desired accuracy is achieved. Here is how you can estimate the roots of f(x) = sin(x) + x^2 to at least 5 digits of accuracy using the Newton-Raphson method:

Calculate the derivative of f(x): f'(x) = cos(x) + 2x
Choose an initial guess, x0.
Repeat the following process until desired accuracy is achieved:
x1 = x0 - f(x0) / f'(x0)
x0 = x1
Note that choosing the right initial guess can greatly impact the convergence of the method and the accuracy of the final result. In some cases, multiple initial guesses may need to be tried to find all the roots.
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