What is the integral of (x^2-14x-5)/((x-1)^2(x^2+1)) ?

The integral of (x^2-14x-5)/((x-1)^2(x^2+1)) is 3ln|x-1|+9/(x-1)-3/2 ln|x^2+1|+7arctan(x)+C

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integral of (x^2-14x-5)/((x-1)^2(x^2+1))

\int \frac{x ^{2} - 14 x - 5}{( x - 1 ) ^{2} ( x ^{2} + 1 )} d x

3 ln ∣ x - 1 ∣ + \frac{9}{x - 1} - \frac{3}{2} ln x^{2} + 1 + 7 arctan (x) + C

Solution steps

\int \frac{x ^{2} - 14 x - 5}{( x - 1 ) ^{2} ( x ^{2} + 1 )} d x

Take the partial fraction of

\frac{x ^{2} - 14 x - 5}{( x - 1 ) ^{2} ( x ^{2} + 1 )} : \frac{3}{x - 1} - \frac{9}{( x - 1 ) ^{2}} + \frac{- 3 x + 7}{x ^{2} + 1}

= \int \frac{3}{x - 1} - \frac{9}{( x - 1 ) ^{2}} + \frac{- 3 x + 7}{x ^{2} + 1} 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{3}{x - 1} d x - \int \frac{9}{( x - 1 ) ^{2}} d x + \int \frac{- 3 x + 7}{x ^{2} + 1} d x

\int \frac{3}{x - 1} d x = 3 ln ∣ x - 1 ∣

\int \frac{9}{( x - 1 ) ^{2}} d x = - \frac{9}{x - 1}

\int \frac{- 3 x + 7}{x ^{2} + 1} d x = - \frac{3}{2} ln x^{2} + 1 + 7 arctan (x)

= 3 ln ∣ x - 1 ∣ - (- \frac{9}{x - 1}) - \frac{3}{2} ln x^{2} + 1 + 7 arctan (x)

Simplify

= 3 ln ∣ x - 1 ∣ + \frac{9}{x - 1} - \frac{3}{2} ln x^{2} + 1 + 7 arctan (x)

Add a constant to the solution

= 3 ln ∣ x - 1 ∣ + \frac{9}{x - 1} - \frac{3}{2} ln x^{2} + 1 + 7 arctan (x) + C

d er i v a t i v e f (x) = 9 x + 5

\int r^{2} - x^{2} d x

\frac{d y}{d x} = 6 - x

t an g e n t f (x) = 2 x^{2} - 10 x + 9, (2, - 3)

(x^{2} + y^{2}) d x + 2 x y d y = 0

What is the integral of (x^2-14x-5)/((x-1)^2(x^2+1)) ?
The integral of (x^2-14x-5)/((x-1)^2(x^2+1)) is 3ln|x-1|+9/(x-1)-3/2 ln|x^2+1|+7arctan(x)+C