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

The integral of (5x 2/3+2)/(3x^2+5x) is 10/9 ln|3x+5|-2/5 (ln| 6/5 x+2|-ln(6/5 |x|))+C

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

integral

\frac{5 x \frac{2}{3} + 2}{3 x ^{2} + 5 x}

\frac{10}{9} ln ∣ 3 x + 5 ∣ - \frac{2}{5} (ln \frac{6}{5} x + 2 - ln (\frac{6}{5} ∣ x ∣)) + C

Solution steps

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

Expand

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

\int \frac{5 x \frac{2}{3}}{3 x ^{2} + 5 x} d x = \frac{10}{9} ln ∣ 3 x + 5 ∣

\int \frac{2}{3 x ^{2} + 5 x} d x = - \frac{2}{5} (ln \frac{6}{5} x + 2 - ln (\frac{6}{5} ∣ x ∣))

= \frac{10}{9} ln ∣ 3 x + 5 ∣ - \frac{2}{5} (ln \frac{6}{5} x + 2 - ln (\frac{6}{5} ∣ x ∣))

Add a constant to the solution

= \frac{10}{9} ln ∣ 3 x + 5 ∣ - \frac{2}{5} (ln \frac{6}{5} x + 2 - ln (\frac{6}{5} ∣ x ∣)) + C

4 x^{2} - 62 x

(x^{2} + 1)^{0.5}

\frac{1}{( x + a )}

(ln (x))^{3}

\frac{2}{( 1 + x ^{2} )}

What is the integral of (5x 2/3+2)/(3x^2+5x) ?
The integral of (5x 2/3+2)/(3x^2+5x) is 10/9 ln|3x+5|-2/5 (ln| 6/5 x+2|-ln(6/5 |x|))+C