Integral Of Sqrt 4 X 2

The Integral of √(4x²): A Step-by-Step Guide to Solving and Understanding

The integral of √(4x²) is a foundational problem in calculus that tests your ability to simplify complex expressions and apply integration techniques. Worth adding: at first glance, the integrand √(4x²) might seem daunting, but breaking it down reveals a straightforward solution. This article will guide you through the process, explain the underlying principles, and address common questions to deepen your understanding.

Step 1: Simplify the Integrand

The first step in solving ∫√(4x²) dx is simplifying the square root term.
√(4x²) = √4 · √(x²) = 2|x|.
Here, the absolute value |x| ensures the result is non-negative, as the square root function always returns a non-negative value. This simplification transforms the integral into:
∫2|x| dx.

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Step 2: Handle the Absolute Value

The absolute value |x| introduces a piecewise nature to the function:

• For x ≥ 0, |x| = x.
• For x < 0, |x| = -x.

This means the integral splits into two cases:

1. Think about it: When x ≥ 0: ∫2x dx = x² + C. 2.