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integral of (10)/((x-1)(x^2+9))

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∫ 10(x−1)(x²+9) dx

Solution

ln|x−1|−12 ln|x²+9|−13 arctan(x3 )+C

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∫ 10(x−1)(x²+9) dx

Take the constant out: ∫a·f(x)dx=a·∫f(x)dx

=10· ∫ 1(x−1)(x²+9) dx

Take the partial fraction of 1(x−1)(x²+9) : 110(x−1) +−x−110(x²+9)

=10· ∫ 110(x−1) +−x−110(x²+9) dx

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

=10(∫ 110(x−1) dx+∫ −x−110(x²+9) dx)

∫ 110(x−1) dx=110 ln|x−1|

∫ −x−110(x²+9) dx=110 (−12 ln|x²+9|−13 arctan(x3 ))

=10(110 ln|x−1|+110 (−12 ln|x²+9|−13 arctan(x3 )))

Simplify 10(110 ln|x−1|+110 (−12 ln|x²+9|−13 arctan(x3 ))): ln|x−1|−12 ln|x²+9|−13 arctan(x3 )

=ln|x−1|−12 ln|x²+9|−13 arctan(x3 )

Add a constant to the solution

=ln|x−1|−12 ln|x²+9|−13 arctan(x3 )+C

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

integral of (10)/((x-1)(x^2+9))

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