`int|x|^3\ dx` is equal to `( x^4)/4+C` (b) `(|x|^4)/4+C` (c) `(x^4)/4+C` (d) none of these

Explanation

To solve the integral \( \int |x|^3 \, dx \), we need to consider the definition of the absolute value function, which behaves differently based on whether \( x \) is positive or negative.

### Step 1: Split the integral based on the sign of \( x \)

The absolute value function \( |x| \) can be defined as:
- \( |x| = x \) when \( x \geq 0 \)
- \( |x| = -x \) when \( x < 0 \)

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