Find the integral ∫ (x + sin(x))^2 dx. Which part is included in the result?

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To find the integral of ( \int (x + \sin(x))^2 dx ), we will first expand the integrand:

[

(x + \sin(x))^2 = x^2 + 2x\sin(x) + \sin^2(x)

]

This allows us to break down the integral into simpler parts:

[

\int (x + \sin(x))^2 dx = \int (x^2 + 2x\sin(x) + \sin^2(x)) dx

]

[

= \int x^2 dx + \int 2x\sin(x) dx + \int \sin^2(x) dx

]

Calculating each integral separately:

For ( \int x^2 dx ), the result is ( \frac{1}{3}x^3 ).
The term ( \int 2x\sin(x) dx ) will be solved using integration by parts.
The integral ( \int \sin^2(x) dx ) can also be calculated, often utilizing the power-reduction formula.

The correct answer includes the results of these integrals and specifically mentions the integration of (