If f’(x) = 8x3 – 2x, f(2) = 8, find f(x).

1.	If f’(x) = 8x3 – 2x, f(2) = 8, find f(x).
Answer» Given, f’(x) = 8x³ – 2x and f(2) = 8 On integrating the given equation, we have \(\int\)f'(x)dx = \(\int\)(8x³-2x) dx We know, \(\int\)f'(x)dx = f(x) ⇒ f(x) = \(\int\)(8x³-2x) dx ⇒ f(x) = \(\int\)(8x³-2x) dx - \(\int\)2xdx ⇒ f(x) = 8\(\int\)(x³dx - 2\(\int\)xdx Recall \(\int\)xⁿdx = \(\frac{x^{n+1}}{n+1}\) + c ⇒ f(x) = 8(\(\frac{x^{3+1}}{3+1}\)) - 2(\(\frac{x^{1+1}}{1+1}\)) + c ⇒ f(x) = 8(\(\frac{x^{4}}{4}\)) - 2(\(\frac{x^{2}}{2}\)) + c ⇒ f(x) = 2x⁴ – x² + c On substituting x = 2 in f(x), we get f(2) = 2(2⁴) – 2² + c ⇒ 8 = 32 – 4 + c ⇒ 8 = 28 + c ∴ c = –20 On substituting the value of c in f(x), we get f(x) = 2x⁴ – x² + (–20) ∴ f(x) = 2x⁴– x² – 20 Thus, f(x) = 2x⁴ – x² – 20

Answer»

Given,

f’(x) = 8x³ – 2x and f(2) = 8

On integrating the given equation, we have

\(\int\)f'(x)dx = \(\int\)(8x³-2x) dx

We know,

\(\int\)f'(x)dx = f(x)

⇒ f(x) = \(\int\)(8x³-2x) dx

⇒ f(x) = \(\int\)(8x³-2x) dx - \(\int\)2xdx

⇒ f(x) = 8\(\int\)(x³dx - 2\(\int\)xdx

Recall \(\int\)xⁿdx = \(\frac{x^{n+1}}{n+1}\) + c

⇒ f(x) = 8(\(\frac{x^{3+1}}{3+1}\)) - 2(\(\frac{x^{1+1}}{1+1}\)) + c

⇒ f(x) = 8(\(\frac{x^{4}}{4}\)) - 2(\(\frac{x^{2}}{2}\)) + c

⇒ f(x) = 2x⁴ – x² + c

On substituting x = 2 in f(x), we get

f(2) = 2(2⁴) – 2² + c

⇒ 8 = 32 – 4 + c

⇒ 8 = 28 + c

∴ c = –20

On substituting the value of c in f(x), we get

f(x) = 2x⁴ – x² + (–20)

∴ f(x) = 2x⁴– x² – 20

Thus,

f(x) = 2x⁴ – x² – 20

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