Evaluate $\int$(cos4x + sin3x)dx

Analysis

To solve this problem, we need to apply the rules of integration for trigonometric functions. Specifically, we are integrating the sum of two functions: $\cos(4x)$ and $\sin(3x)$ with respect to $x$.

Concept

The standard integration formulas for trigonometric functions of the form $ax$ are:

1. $\int \cos(ax) \, dx = \frac{1}{a} \sin(ax) + k$
2. $\int \sin(ax) \, dx = -\frac{1}{a} \cos(ax) + k$

Where $k$ is the constant of integration.

Step-by-Step Solution

Step 1: Split the integral We can integrate each term separately:

Step 2: Integrate $\cos 4x$ Using the formula $\int \cos(ax) \, dx = \frac{1}{a} \sin(ax)$:

Step 3: Integrate $\sin 3x$ Using the formula $\int \sin(ax) \, dx = -\frac{1}{a} \cos(ax)$:

Step 4: Combine the results and add the constant $k$

Comparing this result with the given options, it matches option C.

Exam Tip

When integrating trigonometric functions with a coefficient inside the bracket (like $4x$ or $3x$), always remember to divide by that coefficient. A common mistake students make is multiplying by the coefficient instead of dividing. Think of integration as the "reverse" of differentiation—since differentiation involves multiplying by the coefficient (Chain Rule), integration must involve dividing by it!