The integral of x + sin(x) + x*sin(x) can be found by integrating each term separately.

The integral of x with respect to x is (1/2)*x^2 + C, where C is the constant of integration.

The integral of sin(x) with respect to x is -cos(x) + C.

To integrate x*sin(x), we can use integration by parts. Let u = x and dv/dx = sin(x), then du/dx = 1 and v = -cos(x). Therefore,

integral of xsin(x) dx = -xcos(x) + integral of cos(x) dx
= -x*cos(x) + sin(x) + C

Putting all of the integrals together, we get:

integral of x + sin(x) + x*sin(x) dx = (1/2)x^2 - xcos(x) + sin(x) + C

Therefore, (1/2)x^2 - xcos(x) + sin(x) + C is the antiderivative of x + sin(x) + x*sin(x).

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