What is the integral of (x^2+1)/((x-6)(x-5)^2) ?

The integral of (x^2+1)/((x-6)(x-5)^2) is 37ln|x-6|-36ln|x-5|+(26)/(x-5)+C

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integral of (x^2+1)/((x-6)(x-5)^2)

\int \frac{x ^{2} + 1}{( x - 6 ) ( x - 5 ) ^{2}} d x

37 ln ∣ x - 6 ∣ - 36 ln ∣ x - 5 ∣ + \frac{26}{x - 5} + C

Solution steps

\int \frac{x ^{2} + 1}{( x - 6 ) ( x - 5 ) ^{2}} d x

Take the partial fraction of

\frac{x ^{2} + 1}{( x - 6 ) ( x - 5 ) ^{2}} : \frac{37}{x - 6} - \frac{36}{x - 5} - \frac{26}{( x - 5 ) ^{2}}

= \int \frac{37}{x - 6} - \frac{36}{x - 5} - \frac{26}{( x - 5 ) ^{2}} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{37}{x - 6} d x - \int \frac{36}{x - 5} d x - \int \frac{26}{( x - 5 ) ^{2}} d x

\int \frac{37}{x - 6} d x = 37 ln ∣ x - 6 ∣

\int \frac{36}{x - 5} d x = 36 ln ∣ x - 5 ∣

\int \frac{26}{( x - 5 ) ^{2}} d x = - \frac{26}{x - 5}

= 37 ln ∣ x - 6 ∣ - 36 ln ∣ x - 5 ∣ - (- \frac{26}{x - 5})

Simplify

= 37 ln ∣ x - 6 ∣ - 36 ln ∣ x - 5 ∣ + \frac{26}{x - 5}

Add a constant to the solution

= 37 ln ∣ x - 6 ∣ - 36 ln ∣ x - 5 ∣ + \frac{26}{x - 5} + C

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x \to 0 lim (x^{\frac{1}{l n ( x )}})

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What is the integral of (x^2+1)/((x-6)(x-5)^2) ?
The integral of (x^2+1)/((x-6)(x-5)^2) is 37ln|x-6|-36ln|x-5|+(26)/(x-5)+C