Evaluate: $\int x \, dx$

Question

The derivative of $\sin x$ is:

Answer 1

To evaluate the integral $\int x \, dx$, we follow the fundamental rules of integration.

The integral of $x^n$ with respect to $x$ is given by the formula:

\[\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\]

where $n \neq -1$, and $C$ is the constant of integration.

In this case, we have $n = 1$. Substitute this value into the formula:

\[\int x \, dx = \frac{x^{1+1}}{1+1} + C = \frac{x^2}{2} + C\]

Thus, the solution to the integral is:

\[\frac{x^2}{2} + C\]

Correct Answer: $\frac{x^2}{2} + C$

Let's eliminate the other options:

$x$: This would be the derivative of $\frac{x^2}{2} + C$, not the integral.
$\ln x$: The integral of $\ln x$ is $x \ln x - x + C$, not the integral of $x$.
$x^2 + C$: This is incorrect because it lacks the division by 2, which is necessary as per the formula for integration.

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