What is the integral of (2X^2-7X-2)/(X^3-X) ?

The integral of (2X^2-7X-2)/(X^3-X) is 5/2 ln|X+1|-9/2 ln|X-1|+ln|X^2-1|+2ln|X|+C

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integral of (2X^2-7X-2)/(X^3-X)

\int \frac{2 X ^{2} - 7 X - 2}{X ^{3} - X} d X

\frac{5}{2} ln ∣ X + 1 ∣ - \frac{9}{2} ln ∣ X - 1 ∣ + ln X^{2} - 1 + 2 ln ∣ X ∣ + C

Solution steps

\int \frac{2 X ^{2} - 7 X - 2}{X ^{3} - X} d X

Expand

\frac{2 X ^{2} - 7 X - 2}{X ^{3} - X} : \frac{2 X ^{2}}{X ^{3} - X} - \frac{7 X}{X ^{3} - X} - \frac{2}{X ^{3} - 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{2 X ^{2}}{X ^{3} - X} d X - \int \frac{7 X}{X ^{3} - X} d X - \int \frac{2}{X ^{3} - X} d X

\int \frac{2 X ^{2}}{X ^{3} - X} d X = ln X^{2} - 1

\int \frac{7 X}{X ^{3} - X} d X = - 7 (\frac{1}{2} ln ∣ X + 1 ∣ - \frac{1}{2} ln ∣ X - 1 ∣)

\int \frac{2}{X ^{3} - X} d X = - 2 ln ∣ X ∣ + ln ∣ X + 1 ∣ + ln ∣ X - 1 ∣

= ln X^{2} - 1 - (- 7 (\frac{1}{2} ln ∣ X + 1 ∣ - \frac{1}{2} ln ∣ X - 1 ∣)) - (- 2 ln ∣ X ∣ + ln ∣ X + 1 ∣ + ln ∣ X - 1 ∣)

Simplify

ln X^{2} - 1 - (- 7 (\frac{1}{2} ln ∣ X + 1 ∣ - \frac{1}{2} ln ∣ X - 1 ∣)) - (- 2 ln ∣ X ∣ + ln ∣ X + 1 ∣ + ln ∣ X - 1 ∣) : \frac{5}{2} ln ∣ X + 1 ∣ - \frac{9}{2} ln ∣ X - 1 ∣ + ln X^{2} - 1 + 2 ln ∣ X ∣

= \frac{5}{2} ln ∣ X + 1 ∣ - \frac{9}{2} ln ∣ X - 1 ∣ + ln X^{2} - 1 + 2 ln ∣ X ∣

Add a constant to the solution

= \frac{5}{2} ln ∣ X + 1 ∣ - \frac{9}{2} ln ∣ X - 1 ∣ + ln X^{2} - 1 + 2 ln ∣ X ∣ + C

y^{^{'}} = t^{0.5}, y (0) = 0

\frac{d}{d x} (5 x e^{3 x})

\int_{0}^{\frac{π}{3}} 8 sec^{3} (x) d x

f (x) = 7 x + 2, (2, 4)

x y^{^{'}} + y = e^{x}, y (1) = 1

What is the integral of (2X^2-7X-2)/(X^3-X) ?
The integral of (2X^2-7X-2)/(X^3-X) is 5/2 ln|X+1|-9/2 ln|X-1|+ln|X^2-1|+2ln|X|+C