Find $\int$ cos4 x dx

To solve the integral $\int \cos(4x) \, dx$, we use the standard rule for integrating trigonometric functions of the form $\cos(ax)$.

1. Analyze the question

We are asked to find the indefinite integral of the function $f(x) = \cos(4x)$ with respect to $x$.

2. Explain the concept

The integration rule for a cosine function with a linear argument $ax$ is:

where $k$ is the constant of integration. This is derived from the reverse of the chain rule in differentiation. Since the derivative of $\sin(4x)$ is $4\cos(4x)$, we must divide by $4$ to balance the equation when integrating $\cos(4x)$.

3. Show the solution

• Step 1: Identify the constant $a$ in the expression $\cos(4x)$. Here, $a = 4$.
• Step 2: Apply the integration formula: $$\int \cos(4x) \, dx = \frac{1}{4} \sin(4x) + k$$
• Step 3: Compare this result with the given options.

4. Identify the answer

The result $\frac{1}{4} \sin(4x) + k$ matches option D.

5. Exam Tip

Whenever you are integrating a trigonometric function like $\sin(ax)$ or $\cos(ax)$, always remember to divide by the coefficient of $x$. A common mistake is to multiply by the coefficient instead of dividing. Always check your answer by differentiating it; if you differentiate $\frac{1}{4} \sin(4x)$, you get $\frac{1}{4} \cdot 4 \cos(4x) = \cos(4x)$, which confirms the answer is correct.