Integrate y = 4x\(^3\) + 2x + cos x.

To integrate the function y = 4x^3 + 2x + \cos x, we need to find the indefinite integral (also called the antiderivative) for each term separately and then sum the results, remembering to include the constant of integration, \( C \).

• Integral of \( 4x^3 \):
  Recall that the integral of \( x^n \) (where n ≠ -1) is \( \frac{x^{n+1}}{n+1} \).
  \[ \int 4x^3 \, dx = 4 \int x^3 \, dx = 4 \left( \frac{x^{4}}{4} \right) = x^4 \]
• Integral of \( 2x \):
  \[ \int 2x \, dx = 2 \int x \, dx = 2 \left( \frac{x^{2}}{2} \right) = x^2 \]
• Integral of \( \cos x \):
  The antiderivative of \( \cos x \) is \( \sin x \).
  \[ \int \cos x \, dx = \sin x \]

Now, sum all the results and add the arbitrary constant \( C \):

This result shows that the correct answer is the one where the antiderivative is \( x^4 + x^2 + \sin x + C \).

Notice:

• The sign in front of sin x must be positive since the integral of \( \cos x \) is \( \sin x \), not \( -\sin x \).
• There are no minus signs in front of \( x^2 \) or \( x^4 \) because integrating \( 4x^3 \) and \( 2x \) both yield positive powers as shown.

Key concept: When integrating, apply the power rule for terms with \( x \) and use standard integral rules for trigonometric functions. Always check the sign and degree of each term after integrating.