1. The problem is to find the indefinite integral of the function $3x^{37}$ with respect to $x$. 2. The formula for integrating a power function $x^n$ is: $$\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$$ where $n \neq -1$ and $C$ is the constant of integration. 3. Applying this formula to $3x^{37}$, we treat the constant 3 as a multiplier: $$\int 3x^{37} \, dx = 3 \int x^{37} \, dx$$ 4. Using the power rule: $$3 \int x^{37} \, dx = 3 \cdot \frac{x^{37+1}}{37+1} + C = 3 \cdot \frac{x^{38}}{38} + C$$ 5. Simplify the expression: $$\frac{3}{38} x^{38} + C$$ 6. Therefore, the integral of $3x^{37}$ with respect to $x$ is: $$\boxed{\frac{3}{38} x^{38} + C}$$