\( \int _{}^{}2x d \)

To evaluate the integral \( \int 2x \, dx \), we can apply the power rule of integration. When you integrate a term \( ax^n \), the result is \( \frac{a}{n+1} x^{n+1} + C \) where \( C \) is the constant of integration. Therefore, for \( 2x \), we increase the exponent by 1, which gives us \( 2 \cdot \frac{x^{1+1}}{1+1} = \frac{2}{2} x^2 = x^2 \). So, the final answer is \( x^2 + C \). In real-world terms, integrals like this one can represent areas under curves, which is essential in fields like physics and economics. For example, calculating the total distance traveled given a velocityfunction involves integrating the velocity with respect to time. So understanding these concepts can be crucial for interpreting and solving real-life problems!