What is the integral of (2x+5)/(3x(x+3)^2) ?

The integral of (2x+5)/(3x(x+3)^2) is

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integral of (2x+5)/(3x(x+3)^2)

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

\frac{5 x ^{2} ln ∣ x ∣ - 5 x ^{2} ln ∣ x + 3 ∣ + 30 x ln ∣ x ∣ - 3 x - 30 x ln ∣ x + 3 ∣ + 45 ln ∣ x ∣ - 45 ln ∣ x + 3 ∣ - 9}{27 ( x + 3 ) ^{2}} + C

Solution steps

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

Expand

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

Apply the Sum Rule:

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

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

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

\int \frac{5}{x ( x + 3 ) ^{2}} d x = 5 (\frac{1}{9} ln ∣ x ∣ - \frac{1}{9} ln ∣ x + 3 ∣ + \frac{1}{3 ( x + 3 )})

= \frac{1}{3} (- \frac{2}{x + 3} + 5 (\frac{1}{9} ln ∣ x ∣ - \frac{1}{9} ln ∣ x + 3 ∣ + \frac{1}{3 ( x + 3 )}))

Simplify

\frac{1}{3} (- \frac{2}{x + 3} + 5 (\frac{1}{9} ln ∣ x ∣ - \frac{1}{9} ln ∣ x + 3 ∣ + \frac{1}{3 ( x + 3 )})) : \frac{5 x ^{2} ln ∣ x ∣ - 5 x ^{2} ln ∣ x + 3 ∣ + 30 x ln ∣ x ∣ - 3 x - 30 x ln ∣ x + 3 ∣ + 45 ln ∣ x ∣ - 45 ln ∣ x + 3 ∣ - 9}{27 ( x + 3 ) ^{2}}

= \frac{5 x ^{2} ln ∣ x ∣ - 5 x ^{2} ln ∣ x + 3 ∣ + 30 x ln ∣ x ∣ - 3 x - 30 x ln ∣ x + 3 ∣ + 45 ln ∣ x ∣ - 45 ln ∣ x + 3 ∣ - 9}{27 ( x + 3 ) ^{2}}

Add a constant to the solution

= \frac{5 x ^{2} ln ∣ x ∣ - 5 x ^{2} ln ∣ x + 3 ∣ + 30 x ln ∣ x ∣ - 3 x - 30 x ln ∣ x + 3 ∣ + 45 ln ∣ x ∣ - 45 ln ∣ x + 3 ∣ - 9}{27 ( x + 3 ) ^{2}} + C

\frac{\partial}{\partial y} (x y + \frac{8}{x} + \frac{8}{y})

\frac{d}{d x} (7 - 2 x)

x \to 0 lim (x cot (\frac{7}{11} x))

\int_{1}^{e} x^{2} ln (x) d x

\frac{d y}{d x} = (x y^{2})

What is the integral of (2x+5)/(3x(x+3)^2) ?
The integral of (2x+5)/(3x(x+3)^2) is