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

The integral of (2x)/(5x+3) is 2(1/5 x-3/25 ln|5x+3|)+C

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

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

2 (\frac{1}{5} x - \frac{3}{25} ln ∣ 5 x + 3 ∣) + C

Solution steps

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

Take the constant out:

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

= 2 \cdot \int \frac{x}{5 x + 3} d x

Long division

\frac{x}{5 x + 3} : \frac{1}{5} - \frac{3}{5 ( 5 x + 3 )}

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

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

\int \frac{1}{5} d x = \frac{1}{5} x

\int \frac{3}{5 ( 5 x + 3 )} d x = \frac{3}{25} ln ∣ 5 x + 3 ∣

= 2 (\frac{1}{5} x - \frac{3}{25} ln ∣ 5 x + 3 ∣)

Add a constant to the solution

= 2 (\frac{1}{5} x - \frac{3}{25} ln ∣ 5 x + 3 ∣) + C

\int e^{a x} sin (x) d x

\int a^{3 x} d x

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

\int \frac{1}{e ^{2 x} - 25} d x

What is the integral of (2x)/(5x+3) ?
The integral of (2x)/(5x+3) is 2(1/5 x-3/25 ln|5x+3|)+C