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When faced with an integral such as \( \int (3 \sin x - 2 \sec^2 x) \, dx \), one can effectively employ the **linear combination technique**. This approach involves decomposing a complicated integral into simpler parts that can be easily tackled individually.
Instead of integrating the entire expression at once, split it into two separate integrals:

• \( \int 3 \sin x \, dx \)
• \( \int 2 \sec^2 x \, dx \)

This method not only simplifies computations but also allows the utilization of known basic integrals. Both trigonometric functions \( \sin x \) and \( \sec^2 x \) have standard integral results, such as \( \int \sin x \, dx = -\cos x \) and \( \int \sec^2 x \, dx = \tan x \).
By applying basic integration rules and constants appropriately, you arrive at the final answer: \(-3 \cos x - 2 \tan x + C\), where \(C\) represents the constant of integration. Splitting up integrals and dealing with each portion separately is a powerful tactic, especially in more complex integrals or where elementary results are readily available.

**Trigonometric integrals** often appear intimidating, but they're manageable with a solid grasp of the basic ideas. In our example, we are dealing with \( \int 3 \sin x \, dx \) and \( \int 2 \sec^2 x \, dx \). Understanding these allows us to quickly apply integrations.

• For \( \int \sin x \, dx \): The integral of \( \sin x \) is \(-\cos x\). For any constant factor, simply multiply it after integrating. Hence, \( \int 3 \sin x \, dx = 3(-\cos x) = -3 \cos x \).
• For \( \int \sec^2 x \, dx \): The integral of \( \sec^2 x \) is \( \tan x \). Again, applying the constant factor here gives \( \int 2 \sec^2 x \, dx = 2 \tan x \).

The presence of constants does not complicate the process but serves as a simple multiplication after performing the integral. Being comfortable with these basic integrations for trigonometric functions is crucial as these fundamental forms appear frequently across calculus problems.

After finding an integral, verifying the result is essential to ensure correctness using **differentiation**. This verification step involves differentiating the result to see if we obtain the original function inside the integral.

• Our result was \(-3 \cos x - 2 \tan x + C\).
• Differentiate both components individually.

First, the derivative of \(-3 \cos x\) is \(3 \sin x\). Using the known derivative \(\frac{d}{dx}(-\cos x) = \sin x\), putting multiplication by constant factor gives \(3\sin x\).
Next, the derivative of \(-2 \tan x\) results in \(-2 \sec^2 x\). This is based on \(\frac{d}{dx}(\tan x) = \sec^2 x\), maintaining the constant factor as \(-2\).
Finally, a constant \(C\) has a derivative of 0.
Combining these, \(3 \sin x - 2 \sec^2 x\) recovers our original function \(3 \sin x - 2 \sec^2 x\), confirming the integration was performed correctly. Verification by differentiation is a vital cross-checking mechanism in calculus that reinforces the accuracy of an integral solution.