Question

Determine the antiderivative F of “f” , which is defined by f (x) = 4x3 – 6, where F (0) = 3​
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Answer 1

Step-by-step explanation:

ɢⁱᵛᵉⁿ ᶠᵘⁿᶜᵗⁱᵒⁿ: ᶠ (ˣ) = 4ˣ3 – 6

ɴᵒʷ, ⁱⁿᵗᵉᵍʳᵃᵗᵉ ᵗʰᵉ ᶠᵘⁿᶜᵗⁱᵒⁿ:

∫4ˣ3 – 6 = 4(ˣ4/4)-6ˣ + ᴄ

∫4ˣ3 – 6 = ˣ4 – 6ˣ + ᴄ

ᴛʰᵘˢ, ᵗʰᵉ ᵃⁿᵗⁱᵈᵉʳⁱᵛᵃᵗⁱᵛᵉ ᵒᶠ ᵗʰᵉ ᶠᵘⁿᶜᵗⁱᵒⁿ, ғ ⁱˢ ˣ4 – 6ˣ + ᴄ, ʷʰᵉʳᵉ ᴄ ⁱˢ ᵃ ᶜᵒⁿˢᵗᵃⁿᵗ

ᴀˡˢᵒ, ᵍⁱᵛᵉⁿ ᵗʰᵃᵗ, ғ(0) = 3,

ɴᵒʷ, ˢᵘᵇˢᵗⁱᵗᵘᵗᵉ ˣ = 0 ⁱⁿ ᵗʰᵉ ᵒᵇᵗᵃⁱⁿᵉᵈ ᵃⁿᵗⁱᵈᵉʳⁱᵛᵃᵗⁱᵛᵉ ᶠᵘⁿᶜᵗⁱᵒⁿ, ʷᵉ ᵍᵉᵗ:

(0)4 – 6(0) + ᴄ = 3

ᴛʰᵉʳᵉᶠᵒʳᵉ, ᴄ = 3.

ɴᵒʷ, ˢᵘᵇˢᵗⁱᵗᵘᵗᵉ ᴄ = 3 ⁱⁿ ᵃⁿᵗⁱᵈᵉʳⁱᵛᵃᵗⁱᵛᵉ ᶠᵘⁿᶜᵗⁱᵒⁿ

ʜᵉⁿᶜᵉ, ᵗʰᵉ ʳᵉᵠᵘⁱʳᵉᵈ ᵃⁿᵗⁱᵈᵉʳⁱᵛᵃᵗⁱᵛᵉ ᶠᵘⁿᶜᵗⁱᵒⁿ ⁱˢ ˣ4 – 6ˣ + 3

ʜᵒᵖᵉ ⁱᵗ'ˢ ʰᵉˡᵖ ᵘʰ ❤️.....

Answer 2

Given Function :

ᶠ (ˣ) = 4ˣ3 – 6

ɴow,

integrate the function..

∫4ˣ3 – 6 = 4(ˣ4/4)-6ˣ + ᴄ ∫4ˣ3 – 6 = ˣ4 – 6ˣ + ᴄ

ᴛhus,

the antiderivative function is the ,

ғ ⁱˢ ˣ4 – 6ˣ + ᴄ,

where ᴄ is a constant

ᴀlso, given that , ғ(0) = 3,

ɴow, substitute ˣ = 0

in the antiderivative function ,

we get ,

(0)4 – 6(0) + ᴄ = 3

ᴛherefore, ᴄ = 3.

ɴow, substitute ᴄ = 3 ⁱⁿ antiderivative function

Hence, the required antiderivative function isˣ4 – 6ˣ + 3

ʜᵒᵖᵉ ⁱᵗ'ˢ ʰᵉˡᵖ ᵘʰ ❤️

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