Integral Of 1 Sqrt 4 X 2 Solved Step By Step Clearly
Integral of 1/sqrt(4x^2) and the key transformation
The integral of 1/sqrt(4x^2) is best handled by rewriting the square root first: $$\sqrt{4x^2}=2|x|$$, so the integrand becomes $$\frac{1}{2|x|}$$, not a standard single-form antiderivative across all real $$x$$ .
Core transformation
The key transformation is to recognize that $$\sqrt{4x^2}$$ simplifies to an absolute-value expression, which means the integral depends on the sign of $$x$$ . In practice, this turns a seemingly algebraic radical into a piecewise logarithmic form, because $$\int \frac{1}{x}\,dx=\ln|x|+C$$ is the underlying pattern .
Antiderivative
For $$x>0$$, $$\int \frac{1}{\sqrt{4x^2}}\,dx=\int \frac{1}{2x}\,dx=\frac{1}{2}\ln|x|+C$$ . For $$x<0$$, the same simplification gives $$\int \frac{1}{2|x|}\,dx=-\frac{1}{2}\ln|x|+C$$, so the cleanest answer is to state the result piecewise .
Worked result
When a calculus problem is written in this shorthand form, the safest interpretation is usually the real-domain integral of $$\frac{1}{\sqrt{4x^2}}$$, which equals $$\frac{1}{2|x|}$$ before integration . That means the antiderivative cannot be expressed as one globally smooth formula without noting the domain split at $$x=0$$ .
| Expression | Simplified form | Antiderivative |
|---|---|---|
| $$\frac{1}{\sqrt{4x^2}}$$ | $$\frac{1}{2|x|}$$ | Piecewise, depending on the sign of $$x$$ |
| $$\frac{1}{2x}$$ for $$x>0$$ | $$\frac{1}{2x}$$ | $$\frac{1}{2}\ln|x|+C$$ |
| $$\frac{1}{2|x|}$$ for $$x<0$$ | $$-\frac{1}{2x}$$ | $$-\frac{1}{2}\ln|x|+C$$ |
Step-by-step method
- Rewrite the radical as $$\sqrt{4x^2}=2|x|$$ .
- Convert the integrand to $$\frac{1}{2|x|}$$ .
- Split into cases $$x>0$$ and $$x<0$$ if you need a real antiderivative.
- Integrate the resulting reciprocal form using the logarithm rule.
Why this matters in class
This problem is a good example of why students should always simplify radicals before choosing a technique, because the algebraic structure determines the calculus method. In a Marist classroom setting, that habit reinforces precision, patience, and disciplined reasoning, which are essential for both mathematics learning and academic formation.
Common mistakes
- Forgetting the absolute value in $$\sqrt{4x^2}=2|x|$$ .
- Treating the integrand as $$\frac{1}{2x}$$ without stating the domain.
- Using a trigonometric substitution when simple algebra already resolves the expression.
- Giving one antiderivative for all real $$x$$ without a piecewise explanation.
"Simplify first, then integrate." That is the most reliable transformation rule for problems of this type, especially when radicals hide absolute values and domain issues.
Everything you need to know about Integral Of 1 Sqrt 4 X 2 Solved Step By Step Clearly
What is the integral of 1/sqrt(4x^2)?
It simplifies to the integral of $$\frac{1}{2|x|}$$, so the antiderivative is piecewise and depends on whether $$x$$ is positive or negative .
Why does the absolute value appear?
Because $$\sqrt{x^2}=|x|$$ for real numbers, and therefore $$\sqrt{4x^2}=2|x|$$ .
Is there a single formula for all x?
Not as a real-valued elementary antiderivative without noting the sign of $$x$$, since the function changes behavior at $$x=0$$.
