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integral of ((x^3+9))/(x^2+3x+2)

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∫ (x³+9)x²+3x+2 dx

Solution

x²2 −3x+8ln|x+1|−ln|x+2|+C

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∫ x³+9x²+3x+2 dx

Long division x³+9x²+3x+2 : x−3+7x+15x²+3x+2

=∫ x−3+7x+15x²+3x+2 dx

Apply the Sum Rule: ∫f(x)±g(x)dx=∫f(x)dx±∫g(x)dx

=∫ xdx−∫ 3dx+∫ 7x+15x²+3x+2 dx

∫ xdx=x²2

∫ 3dx=3x

∫ 7x+15x²+3x+2 dx=8ln|x+1|−ln|x+2|

=x²2 −3x+8ln|x+1|−ln|x+2|

Add a constant to the solution

=x²2 −3x+8ln|x+1|−ln|x+2|+C

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−▭▭	<	7	8	9	÷	`AC`
+▭▭	>	4	5	6	×	☐☐☐
×▭▭	(	1	2	3	−	`x`
▭ \|▭	)	.	0	=	+	`y`

integral of ((x^3+9))/(x^2+3x+2)

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