Suppose a curve 𝑦(𝑥) has the derivatived𝑦d𝑥=5−2𝑥.1. Find 𝑦 in terms of 𝑥.𝑦(𝑥)=

To find $y$ in terms of $x$ given the derivative $\frac{dy}{dx} = 5 - 2x$, we need to integrate the derivative with respect to $x$.

Here are the steps:

1. Write down the given derivative: $\frac{dy}{dx} = 5 - 2x$

2. Integrate both sides with respect to $x$: $\int \frac{dy}{dx} \, dx = \int (5 - 2x) \, dx$

3. The left side simplifies to $y$: $y = \int (5 - 2x) \, dx$

4. Integrate the right side:

   • The integral of $5$ with respect to $x$ is $5x$.
   • The integral of $-2x$ with respect to $x$ is $-x^2$ (since $\int x \, dx = \frac{x^2}{2}$ and $-2 \times \frac{x^2}{2} = -x^2$).

   So, we have: $y = 5x - x^2 + C$

5. Include the constant of integration $C$: $y = 5x - x^2 + C$

Therefore, the function $y$ in terms of $x$ is: $y(x) = 5x - x^2 + C$

where $C$ is an arbitrary constant.