Find the indefinite integral. (Note: Solve by the most convenient method-not all require integration by parts. Remember the constant of integration.) \( \int x^{2} \cos (x) d x \)

To solve the integral \( \int x^{2} \cos (x) d x \), we can use integration by parts. We set \( u = x^2 \) and \( dv = \cos(x) dx \). Differentiating and integrating, we have \( du = 2x dx \) and \( v = \sin(x) \). Using the integration by parts formula \( \int u \, dv = uv - \int v \, du \): \[ \int x^2 \cos(x) dx = x^2 \sin(x) - \int \sin(x) (2x) dx \] Now, we need to integrate \( 2x \sin(x) dx \) using integration by parts again. Set \( u = 2x \) and \( dv = \sin(x) dx \). Then \( du = 2 dx \) and \( v = -\cos(x) \). So, \[ \int 2x \sin(x) dx = -2x \cos(x) + \int 2 \cos(x) dx = -2x \cos(x) + 2 \sin(x) \] Substituting this back into our first integral, we have: \[ \int x^2 \cos(x) dx = x^2 \sin(x) + 2x \cos(x) - 2\sin(x) + C \] Thus, the final result is: \[ \int x^{2} \cos (x) d x = x^2 \sin(x) + 2x \cos(x) - 2\sin(x) + C \]