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

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

• A. (1/3)x^3 - (1/2)x^2 + C + ∫ (2xsin(x) + sin^2(x)) dx
• B. (1/3)x^3 + C
• C. (1/2)x^2 + sin(x) + C
• D. tan(x) + C

Answer: (A)

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: 1. For \( \int x^2 dx \), the result is \( \frac{1}{3}x^3 \). 2. The term \( \int 2x\sin(x) dx \) will be solved using integration by parts. 3. 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 \(

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:

1. For ( \int x^2 dx ), the result is ( \frac{1}{3}x^3 ).

2. The term ( \int 2x\sin(x) dx ) will be solved using integration by parts.

3. 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 (