What is the integral of (24y-14)/(6y^2-7y) ?

The integral of (24y-14)/(6y^2-7y) is 4ln|6y-7|+4(1/2 ln(12/7 |y|)-1/2 ln| 12/7 y-2|)+C

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integral of (24y-14)/(6y^2-7y)

\int \frac{24 y - 14}{6 y ^{2} - 7 y} d y

4 ln ∣ 6 y - 7 ∣ + 4 (\frac{1}{2} ln (\frac{12}{7} ∣ y ∣) - \frac{1}{2} ln \frac{12}{7} y - 2) + C

Solution steps

\int \frac{24 y - 14}{6 y ^{2} - 7 y} d y

Expand

\frac{24 y - 14}{6 y ^{2} - 7 y} : \frac{24 y}{6 y ^{2} - 7 y} - \frac{14}{6 y ^{2} - 7 y}

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{24 y}{6 y ^{2} - 7 y} d y - \int \frac{14}{6 y ^{2} - 7 y} d y

\int \frac{24 y}{6 y ^{2} - 7 y} d y = 4 ln ∣ 6 y - 7 ∣

\int \frac{14}{6 y ^{2} - 7 y} d y = - 4 (\frac{1}{2} ln (\frac{12}{7} ∣ y ∣) - \frac{1}{2} ln \frac{12}{7} y - 2)

= 4 ln ∣ 6 y - 7 ∣ - (- 4 (\frac{1}{2} ln (\frac{12}{7} ∣ y ∣) - \frac{1}{2} ln \frac{12}{7} y - 2))

Simplify

= 4 ln ∣ 6 y - 7 ∣ + 4 (\frac{1}{2} ln (\frac{12}{7} ∣ y ∣) - \frac{1}{2} ln \frac{12}{7} y - 2)

Add a constant to the solution

= 4 ln ∣ 6 y - 7 ∣ + 4 (\frac{1}{2} ln (\frac{12}{7} ∣ y ∣) - \frac{1}{2} ln \frac{12}{7} y - 2) + C

\frac{1}{s ( s + 1 )} - \frac{1}{s + 1}

\int \frac{1}{x ^{2}} 6 + \frac{1}{x} d x

x \to 0 lim (cot (4 x))

\frac{d}{d x} (\frac{2 x ^{2} - 1}{x ^{3} - x})

\frac{1}{2 r} + \frac{1}{9 r ^{(\frac{8}{9})}}

What is the integral of (24y-14)/(6y^2-7y) ?
The integral of (24y-14)/(6y^2-7y) is 4ln|6y-7|+4(1/2 ln(12/7 |y|)-1/2 ln| 12/7 y-2|)+C