6. If (dy)/(dx)=4x^3-4x and y=0 when x=2 find y in terms of x. 7. At the point (x,y) on a curve, the gradient is equal to (2x-1) If the curve passes through the point (3,4) , what is the equation of the curve? 8. Find-the equation of the curve, gradient 2x^2+3x-1 , which passes through the origin. 19. If (dy)/(dx)=px+2 , where p is a constant, express y as a function of x.given that (d^2y)/(dx^2)=6 and that y=4 when x=0

6. To find y in terms of x, we need to integrate the given derivative with respect to x. Integrating, we get: We are given that when . Substituting these values into the equation, we can solve for the constant C: Therefore, the equation of the curve is .7. We are given that the gradient of the curve at any point is equal to . We can use this information to find the equation of the curve.Let's differentiate the equation of the curve with respect to x: We are given that the curve passes through the point . Substituting these values into the equation, we can solve for the constant of integration: Therefore, the equation of the curve is .8. We are given that the gradient of the curve is , and it passes through the origin .To find the equation of the curve, we need to integrate the given gradient with respect to x: Integrating, we get: Since the curve passes through the origin, we can substitute and to solve for the constant C: Therefore, the equation of the curve is .19. We are given that , where p is a constant, and . We are also given that when .First, let's find the equation of the curve by integrating with respect to x: We are given that when . Substituting these values into the equation, we can solve for the constant C: Now, let's differentiate the equation of the curve with respect to x to find the second derivative: We are given that . Equating this to p, we get: Therefore, the equation of the curve is .