Evaluate the following integral: ∫ e^(2log(x)) dx
Understand the Problem
The question asks for the evaluation of the integral of e^(2*log(x)) with respect to x. To solve this, we will first simplify the expression inside the integral using properties of logarithms and then perform the integration.
Answer
$\frac{x^3}{3} + C$
Answer for screen readers
$I = \frac{x^3}{3} + C$
Steps to Solve
- Simplify the expression using logarithm properties
We use the property $a \log_b x = \log_b x^a$ to rewrite the exponent:
$2 \log x = \log x^2$.
Therefore, the integral becomes:
$I = \int e^{\log x^2} dx$
- Simplify further using the exponential and logarithm relationship
Since $e^{\log x} = x$, we have $e^{\log x^2} = x^2$. Thus, the integral becomes:
$I = \int x^2 dx$
- Integrate the simplified expression
Using the power rule for integration, $\int x^n dx = \frac{x^{n+1}}{n+1} + C$, where $C$ is the constant of integration, we have:
$I = \int x^2 dx = \frac{x^{2+1}}{2+1} + C = \frac{x^3}{3} + C$
$I = \frac{x^3}{3} + C$
More Information
The integral of $e^{2 \log x}$ simplifies to $\frac{x^3}{3} + C$. This is achieved by using the property of logarithms to simplify the exponent and then recognizing the inverse relationship between exponential and logarithmic functions.
Tips
A common mistake is to incorrectly apply the properties of logarithms or to forget the constant of integration. Also, some might not recognize the simplification $e^{\log x^2} = x^2$ directly.
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