integral of (5x^2-8x-13)/(x^2-5)

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∫ 5x²−8x−13x²−5 dx

Solution

5x−6√5 ln|x√5 +1|+6√5 ln|x√5 −1|−4ln|x²−5|+C

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Solution steps

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∫ 5x²−8x−13x²−5 dx

Expand 5x²−8x−13x²−5 : 5x²x²−5 −8xx²−5 −13x²−5

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

=∫ 5x²x²−5 dx−∫ 8xx²−5 dx−∫ 13x²−5 dx

∫ 5x²x²−5 dx=−5√52 ln|x√5 +1|+5√52 ln|x√5 −1|+5x

∫ 8xx²−5 dx=4ln|x²−5|

∫ 13x²−5 dx=−132√5 (ln|x√5 +1|−ln|x√5 −1|)

=−5√52 ln|x√5 +1|+5√52 ln|x√5 −1|+5x−4ln|x²−5|−(−132√5 (ln|x√5 +1|−ln|x√5 −1|))

Simplify −5√52 ln|x√5 +1|+5√52 ln|x√5 −1|+5x−4ln|x²−5|−(−132√5 (ln|x√5 +1|−ln|x√5 −1|)): 5x−6√5 ln|x√5 +1|+6√5 ln|x√5 −1|−4ln|x²−5|

=5x−6√5 ln|x√5 +1|+6√5 ln|x√5 −1|−4ln|x²−5|

Add a constant to the solution

=5x−6√5 ln|x√5 +1|+6√5 ln|x√5 −1|−4ln|x²−5|+C

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