Integrate the function y = 3x\(^2\) + 2x - 5 with respect to x.

Answer Details

To integrate the function \( y = 3x^2 + 2x - 5 \) with respect to \( x \), you need to find the indefinite integral:

\[ \int (3x^2 + 2x - 5)\,dx \]

We can integrate each term separately:

• \( \int 3x^2\,dx \): The integral of \( x^n \) is \( \frac{x^{n+1}}{n+1} \), so:
  \[ \int 3x^2\,dx = 3 \times \frac{x^{2+1}}{2+1} = 3 \times \frac{x^3}{3} = x^3 \]
• \( \int 2x\,dx \): Similarly,
  \[ \int 2x\,dx = 2 \times \frac{x^{1+1}}{1+1} = 2 \times \frac{x^2}{2} = x^2 \]
• \( \int -5\,dx \): The integral of a constant \( a \) with respect to \( x \) is \( a x \), so:
  \[ \int -5\,dx = -5x \]

Now, combine these results and add the constant of integration \( C \):

\[ x^3 + x^2 - 5x + C \]

This is the general antiderivative for the given function. Each term has been integrated correctly using the power rule of integration and the constant of integration \( C \) is included to represent all possible antiderivatives.