Integrate y = 4x$^3$ + 2x + cos x.

Analysis

To solve this problem, we need to find the indefinite integral of the function $y = 4x^3 + 2x + \cos x$. We will apply the basic rules of integration term by term.

Concept: The Power Rule and Trigonometric Integration

1. The Power Rule for Integration: To integrate a term of the form $ax^n$, we use the formula: $$\int ax^n \, dx = \frac{ax^{n+1}}{n+1} + C$$ (where $n \neq -1$)
2. Trigonometric Integration: The integral of $\cos x$ is a standard derivative rule in reverse: $$\int \cos x \, dx = \sin x + C$$

Step-by-Step Solution

We are integrating the expression: $\int (4x^3 + 2x + \cos x) \, dx$

Step 1: Integrate $4x^3$ Using the power rule:

Step 2: Integrate $2x$ Using the power rule (remember that $x$ is $x^1$):

Step 3: Integrate $\cos x$ Using the standard trigonometric integral:

Step 4: Combine the results and add the constant of integration ($C$)

Conclusion

Comparing our result to the given options, we find that our derived expression matches option B.

Exam Tip: Always remember to add the constant of integration ($+ C$) whenever you are solving an indefinite integral. In JAMB, examiners often include options without the "$+ C$" to trick students who forget this fundamental rule!