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Write down the function to be integrated.

Computer Science

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Answer

C\text{C}

Step-by-step solution

5 steps

  1. Write down the function to be integrated.

    We need to integrate y=4x3+2x+cos⁡xy = 4x^3 + 2x + \cos x.
    This means we need to find ∫(4x3+2x+cos⁡x) dx\int (4x^3 + 2x + \cos x) \,dx.

  2. Apply the sum rule for integration, integrating each term separately.

    ∫(4x3+2x+cos⁡x) dx=∫4x3 dx+∫2x dx+∫cos⁡x dx\int (4x^3 + 2x + \cos x) \,dx = \int 4x^3 \,dx + \int 2x \,dx + \int \cos x \,dx

  3. Integrate each term using the power rule ∫xn dx=xn+1n+1+C\int x^n \,dx = \frac{x^{n+1}}{n+1} + C and the standard integral for cos⁡x\cos x.

    For the first term:

    ∫4x3 dx=4⋅x3+13+1=4⋅x44=x4\int 4x^3 \,dx = 4 \cdot \frac{x^{3+1}}{3+1} = 4 \cdot \frac{x^4}{4} = x^4

    For the second term:

    ∫2x dx=2⋅x1+11+1=2⋅x22=x2\int 2x \,dx = 2 \cdot \frac{x^{1+1}}{1+1} = 2 \cdot \frac{x^2}{2} = x^2

    For the third term:

    ∫cos⁡x dx=sin⁡x\int \cos x \,dx = \sin x

  4. Combine the integrated terms and add the constant of integration, CC.

    x4+x2+sin⁡x+Cx^4 + x^2 + \sin x + C

  5. Compare the result with the given options.

    The calculated integral is x4+x2+sin⁡x+Cx^4 + x^2 + \sin x + C, which matches option (C).

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