What is the integral of (8x^2+34x+29)/((x+1)(x+2)^2) ?

The integral of (8x^2+34x+29)/((x+1)(x+2)^2) is 3ln|x+1|+5ln|x+2|-7/(x+2)+C

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integral of (8x^2+34x+29)/((x+1)(x+2)^2)

\int \frac{8 x ^{2} + 34 x + 29}{( x + 1 ) ( x + 2 ) ^{2}} d x

3 ln ∣ x + 1 ∣ + 5 ln ∣ x + 2 ∣ - \frac{7}{x + 2} + C

Solution steps

\int \frac{8 x ^{2} + 34 x + 29}{( x + 1 ) ( x + 2 ) ^{2}} d x

Take the partial fraction of

\frac{8 x ^{2} + 34 x + 29}{( x + 1 ) ( x + 2 ) ^{2}} : \frac{3}{x + 1} + \frac{5}{x + 2} + \frac{7}{( x + 2 ) ^{2}}

= \int \frac{3}{x + 1} + \frac{5}{x + 2} + \frac{7}{( x + 2 ) ^{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{3}{x + 1} d x + \int \frac{5}{x + 2} d x + \int \frac{7}{( x + 2 ) ^{2}} d x

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

\int \frac{5}{x + 2} d x = 5 ln ∣ x + 2 ∣

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

= 3 ln ∣ x + 1 ∣ + 5 ln ∣ x + 2 ∣ - \frac{7}{x + 2}

Add a constant to the solution

= 3 ln ∣ x + 1 ∣ + 5 ln ∣ x + 2 ∣ - \frac{7}{x + 2} + C

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What is the integral of (8x^2+34x+29)/((x+1)(x+2)^2) ?
The integral of (8x^2+34x+29)/((x+1)(x+2)^2) is 3ln|x+1|+5ln|x+2|-7/(x+2)+C