. A curve is such that dy/dx=(2x^2-5). If (3, 8) lies on the curve then the equation of the curve is:?

dy/dx = 2x² - 5 -- (1)

=> By making dy the subject of formula in eqn (1), we have

dy = (2x² - 5)dx -- (2)

=> Apply the integral notation ∫ to both sides in eqn (2), i.e.

 ∫dy = ∫(2x² - 5)dx -- (3)

=> From eqn (3), we integrate thus

y = ∫2x² - ∫5 dx

y = [2x^(2 + 1)]/(2 + 1) - 5x + c

y = 2x³/3 - 5x + c -- (4)

=> Now, if (3, 8) which is in the form: (x, y) lies on the curve (which is the eqn (4)), then the true eqn of the curve is obtained as follows:

-> Substitute 3 & 8 for x & y respectively in eqn (4), i.e.

8 = 2(3)³/3 - 5(3) + c

8 = 2(27)/3 - 15 + c

8 = 2(9) - 15 + c

8 = 18 - 15 + c

8 = 3 + c

8 - 3 = c

c = 5

- > Substitute 5 for c in eqn (4), i.e.

y = 2x³/3 - 5x + 5

Finally, the eqn of the curve at (3, 8) is obtained as y = 2x³/3 - 5x + 5.

dy/dx = 2x² - 5

so, y = ∫ 2x² - 5 dx => (2/3)x³ - 5x + C

Then, when x = 3, y = 8 so,

8 = (2/3)(3)³ - 5(3) + C

i.e. 8 = 18 - 15 + C

=> 8 = 3 + C

so, C = 5

Hence, y = (2/3)x³ - 5x + 5

:)>

dy/dx=(2x^2-5).
Y= ᶴ(2x^2-5)dx
= 2x^3/3 -5x + c
Plug (3,8)
We get 8= 18 – 15+c
C=5
Curve eqn is y= 2x^3/3 -5x +5
3y=2x^3-15x+15

yes

y = ⅔x³ - 5x + c
8 = ⅔(3³) - 5(3) + c
c = 8 - 18 + 15 = 5
y = ⅔x³ - 5x + 5