Integration Of 2x Dx Why Basics Still Matter Deeply
Integration of 2x dx is $$x^2 + C$$, because the power rule of integration says $$\int x^n\,dx = \frac{x^{n+1}}{n+1} + C$$, and here $$\int 2x\,dx = 2\int x\,dx = 2\cdot \frac{x^2}{2} + C = x^2 + C$$.
Why this works
The expression $$2x$$ is a constant multiple of $$x$$, so you can pull the 2 outside the integral and apply the standard power rule to the remaining term. This is the same constant-multiplier logic used throughout basic integral calculus.
- Rule used: Constant multiple rule plus power rule.
- Result: $$\int 2x\,dx = x^2 + C$$.
- Meaning of $$C$$: The constant of integration represents any fixed number added to an indefinite integral.
Step-by-step derivation
- Rewrite the integral as $$2\int x\,dx$$.
- Apply the power rule to $$x^1$$, giving $$\frac{x^2}{2}$$.
- Multiply by 2, which cancels the denominator.
- Attach the constant of integration: $$x^2 + C$$.
Compact formula table
| Expression | Integration | Reason |
|---|---|---|
| $$\int 2x\,dx$$ | $$x^2 + C$$ | Constant multiplier and power rule. |
| $$\int x\,dx$$ | $$\frac{x^2}{2} + C$$ | Power rule with $$n=1$$. |
| $$\int 2x^3\,dx$$ | $$\frac{x^4}{2} + C$$ | Same rule applied term-wise. |
Common student errors
One frequent mistake is to write $$\frac{2x^2}{2}$$ and stop there without simplifying, even though the correct final answer is $$x^2 + C$$. Another mistake is to forget the constant $$C$$, which is required for every indefinite integral.
"Add one to the exponent, then divide by the new exponent" is the practical memory rule behind the power rule.
Frequently asked questions
Takeaway for learners
The safest way to solve integral problems like $$\int 2x\,dx$$ is to separate the constant, apply the power rule, and then simplify. That habit builds accuracy and reduces memorization errors in later calculus work.
Key concerns and solutions for Integration Of 2x Dx Why Basics Still Matter Deeply
Why is the answer not $$2x^2 + C$$?
Because integrating $$x$$ gives $$\frac{x^2}{2}$$, and the leading 2 cancels that denominator, leaving $$x^2 + C$$.
Does this rule work for any constant times $$x$$?
Yes. For any constant $$a$$, $$\int ax\,dx = \frac{ax^2}{2} + C$$, which follows directly from the constant coefficient rule and the power rule.
What is the derivative of $$x^2 + C$$?
The derivative is $$2x$$, which confirms that $$x^2 + C$$ is the correct antiderivative of $$2x$$.
