What is the integral of (4x^2+26x+48)/((x+5)(x+2)^2) ?

The integral of (4x^2+26x+48)/((x+5)(x+2)^2) is 2ln|x+5|+2ln|x+2|-4/(x+2)+C

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integral of (4x^2+26x+48)/((x+5)(x+2)^2)

\int \frac{4 x ^{2} + 26 x + 48}{( x + 5 ) ( x + 2 ) ^{2}} d x

2 ln ∣ x + 5 ∣ + 2 ln ∣ x + 2 ∣ - \frac{4}{x + 2} + C

Solution steps

\int \frac{4 x ^{2} + 26 x + 48}{( x + 5 ) ( x + 2 ) ^{2}} d x

Take the partial fraction of

\frac{4 x ^{2} + 26 x + 48}{( x + 5 ) ( x + 2 ) ^{2}} : \frac{2}{x + 5} + \frac{2}{x + 2} + \frac{4}{( x + 2 ) ^{2}}

= \int \frac{2}{x + 5} + \frac{2}{x + 2} + \frac{4}{( 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{2}{x + 5} d x + \int \frac{2}{x + 2} d x + \int \frac{4}{( x + 2 ) ^{2}} d x

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

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

\int \frac{4}{( x + 2 ) ^{2}} d x = - \frac{4}{x + 2}

= 2 ln ∣ x + 5 ∣ + 2 ln ∣ x + 2 ∣ - \frac{4}{x + 2}

Add a constant to the solution

= 2 ln ∣ x + 5 ∣ + 2 ln ∣ x + 2 ∣ - \frac{4}{x + 2} + C

\frac{d}{d x} (sin (\frac{x}{3}))

\frac{d ^{2}}{d x ^{2}} (x^{2} ln (x))

\frac{3 t}{t ^{2} + 4}

\frac{d}{d x} (x (3 x + 2)^{5})

\int x^{2} - sec^{2} (x) d x

What is the integral of (4x^2+26x+48)/((x+5)(x+2)^2) ?
The integral of (4x^2+26x+48)/((x+5)(x+2)^2) is 2ln|x+5|+2ln|x+2|-4/(x+2)+C