What is the integral of (3x)/((x+5)(2x-1)) ?

The integral of (3x)/((x+5)(2x-1)) is 3(5/11 ln|x+5|+1/22 ln|2x-1|)+C

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integral of (3x)/((x+5)(2x-1))

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

3 (\frac{5}{11} ln ∣ x + 5 ∣ + \frac{1}{22} ln ∣ 2 x - 1 ∣) + C

Solution steps

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

Take the constant out:

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

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

Take the partial fraction of

\frac{x}{( x + 5 ) ( 2 x - 1 )} : \frac{5}{11 ( x + 5 )} + \frac{1}{11 ( 2 x - 1 )}

= 3 \cdot \int \frac{5}{11 ( x + 5 )} + \frac{1}{11 ( 2 x - 1 )} 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

= 3 (\int \frac{5}{11 ( x + 5 )} d x + \int \frac{1}{11 ( 2 x - 1 )} d x)

\int \frac{5}{11 ( x + 5 )} d x = \frac{5}{11} ln ∣ x + 5 ∣

\int \frac{1}{11 ( 2 x - 1 )} d x = \frac{1}{22} ln ∣ 2 x - 1 ∣

= 3 (\frac{5}{11} ln ∣ x + 5 ∣ + \frac{1}{22} ln ∣ 2 x - 1 ∣)

Add a constant to the solution

= 3 (\frac{5}{11} ln ∣ x + 5 ∣ + \frac{1}{22} ln ∣ 2 x - 1 ∣) + C

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What is the integral of (3x)/((x+5)(2x-1)) ?
The integral of (3x)/((x+5)(2x-1)) is 3(5/11 ln|x+5|+1/22 ln|2x-1|)+C