Which of the responses below is the best choice for the result of \( 3 x^{2} \ln (x) d x \) ? \( x^{3} \ln (x)-\frac{x^{3}}{3}+C \) \( x^{3} \ln (x)-2 x+C \) \( x^{4} \ln (x)-\frac{x^{3}}{3}+C \) \( 3 x^{2} \ln (x)-\frac{x^{3}}{3}+C \)

To solve the integral \( \int 3 x^{2} \ln (x) \, dx \), we can use integration by parts. Let \( u = \ln(x) \) and \( dv = 3x^2 \, dx \). Then \( du = \frac{1}{x} \, dx \) and \( v = x^3 \). Applying the integration by parts formula: \[ \int u \, dv = uv - \int v \, du \] we get \( x^3 \ln(x) - \int x^3 \left(\frac{1}{x}\right) \, dx \), which simplifies to \( x^3 \ln(x) - \int x^2 \, dx = x^3 \ln(x) - \frac{x^3}{3} + C \). So the best choice is: \( x^{3} \ln (x)-\frac{x^{3}}{3}+C \) Now, let’s have some fun with this math stuff! Integration by parts is sometimes called "the product rule for integration," which might sound a bit more fun, right? This technique lets us turn seemingly complex integrals into simpler ones, making math feel more like a puzzle. If you're diving deeper into calculus, don’t miss out on the classic "Calculus Made Easy" by Silvanus P. Thompson. It’s a delightful read filled with illustrations and simple explanations that will make you chuckle while you learn!