If - 2x3-3x2-8x-1 -2x-6dx -2x-6-5-7-9 2x-6 f x -C then fx is equal to

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Integration by Parts

If ∫ 2x3+3x...

Question

If $\int (2 x^{3} + 3 x^{2} - 8 x + 1) \sqrt{2 x + 6} d x$
$= \frac{\sqrt{2 x + 6}}{5.7.9} (2 x + 6) f (x) + C$ then f(x) is equal to

x^{3} - 6 x^{2} - 9 l x + 297

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7 x^{3} - 3 x^{2} - 932 x + 597

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70 x^{3} - 45 x^{2} - 396 x + 897

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70 x^{3} - 45 x^{2} - 132 x + 597

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Solution

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\int (2 x^{3} + 3 x^{2} - 8 x + 1) \sqrt{2 x + 6} d x

= \frac{\sqrt{2 x + 6}}{579} (2 x + 6) f (x) + C

f (x)

f (x) = \frac{3}{10} x^{4} - \frac{4}{5} x^{3} - 3 x^{2} + \frac{36}{5} x + 11

f (x) = \frac{x^{4}}{4} + \frac{2}{3} x^{3} - \frac{5}{2} x^{2} - 6 x + 7

f (x) = {\{x (x - 2)\}}^{2}

f (x) = {cot}^{- 1} (\frac{1 - x^{2}}{2 x}) + {cot}^{- 1} (\frac{1 - 3 x^{2}}{3 x - x^{3}}) - {cot}^{- 1} (\frac{1 - 6 x^{2} + x^{4}}{4 x - 4 x^{3}})

F^{'} (x)

f (x) = \sqrt{\frac{2 x + 1}{x^{3} - 3 x^{2} + 2 x}}

f (x) = {\begin{matrix} x^{3} - 3 x + 2 x < 2 x^{3} - 6 x^{2} + 9 x + 2, x \geq 2 \end{matrix}

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