Question

The equation of the curve with passes through the point $(0,1)$ and the slope $3x^2+2x+5$ at any point $(x,y)$ is

• A. $y=3x^3+2x^2+5x+1$
• B. $y=2x^3+3x^2+5x+1$
• C. $y=x^3+x^2+5x+1$
• D. $y=x^3+x^2+5x-1$

Answer 1

The correct option is C

$y=x^3+x^2+5x+1$

Explanation for the correct option:

Finding the equation of the curve:

Given $y=3x^2+2x+5$

Finding the equation of curve by integrating the given equation,

$\int {\left(\frac{dy}{dx}\right)}_{(x,y)}=\int 3x^2+2x+5dx$

$y=x^3+x^2+5x+c$

Now substituting the point $(0,1)\text{where}x=0,y=1$ in the derived equation,

$$\begin{gathered} 1=0+0+0+c \\ c=1 \\ y=x^3+x^2+5x+1 \end{gathered}$$

Hence, option (C) is the correct answer.