The problem asks us to find the value of $l$ if $\int x^2 \, dx = lx + \frac{x^3}{3} + c$, where $c$ is the constant of integration.

1. Problem Description

The problem asks us to find the value of $l$ if $\int x^2 \, dx = lx + \frac{x^3}{3} + c$, where $c$ is the constant of integration.

2. Solution Steps

We are given the equation $\int x^2 \, dx = lx + \frac{x^3}{3} + c$. To solve for $l$, we first evaluate the integral on the left side.

The power rule for integration states that

$\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$, where $n \neq -1$ and $C$ is the constant of integration.

In this case, $n=2$, so

$\int x^2 \, dx = \frac{x^{2+1}}{2+1} + C = \frac{x^3}{3} + C$.

Thus, we have

$\frac{x^3}{3} + C = lx + \frac{x^3}{3} + c$.

Subtracting $\frac{x^3}{3}$ from both sides, we get

$C = lx + c$.

Since $C$ and $c$ are both constants, the only way for this equation to hold for all $x$ is if $l=0$. Otherwise, the right side would depend on $x$.

3. Final Answer

$l = 0$