What is the integral of (5x+7)/(x^2+2x+1) ?

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

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integral of (5x+7)/(x^2+2x+1)

\int \frac{5 x + 7}{x ^{2} + 2 x + 1} d x

5 ln ∣ x + 1 ∣ - \frac{2}{x + 1} + C

Solution steps

\int \frac{5 x + 7}{x ^{2} + 2 x + 1} d x

Complete the square

x^{2} + 2 x + 1 : (x + 1)^{2}

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

Apply u-substitution

: \int \frac{5 u + 2}{u ^{2}} d u

= \int \frac{5 u + 2}{u ^{2}} d u

Expand

\frac{5 u + 2}{u ^{2}} : \frac{5}{u} + \frac{2}{u ^{2}}

= \int \frac{5}{u} + \frac{2}{u ^{2}} d u

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{5}{u} d u + \int \frac{2}{u ^{2}} d u

\int \frac{5}{u} d u = 5 ln ∣ u ∣

\int \frac{2}{u ^{2}} d u = - \frac{2}{u}

= 5 ln ∣ u ∣ - \frac{2}{u}

Substitute back

u = x + 1

= 5 ln ∣ x + 1 ∣ - \frac{2}{x + 1}

Add a constant to the solution

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

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