What is the integral of (2x)/(sqrt(9x^2-6x-8)) ?

The integral of (2x)/(sqrt(9x^2-6x-8)) is 2/9 (3sqrt((x-1/3)^2-1)+ln|sqrt((x-1/3)^2-1)+x-1/3 |)+C

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integral of (2x)/(sqrt(9x^2-6x-8))

\int \frac{2 x}{9 x ^{2} - 6 x - 8} d x

\frac{2}{9} 3 (x - \frac{1}{3})^{2} - 1 + ln (x - \frac{1}{3})^{2} - 1 + x - \frac{1}{3} + C

Solution steps

\int \frac{2 x}{9 x ^{2} - 6 x - 8} 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}{9 x ^{2} - 6 x - 8} d x

Complete the square

9 x^{2} - 6 x - 8 : 9 (x - \frac{1}{3})^{2} - 9

= 2 \cdot \int \frac{x}{9 ( x - \frac{1}{3} ) ^{2} - 9} d x

Apply u-substitution

: \int \frac{3 u + 1}{9 u ^{2} - 1} d u

= 2 \cdot \int \frac{3 u + 1}{9 u ^{2} - 1} d u

Take the constant out:

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

= 2 \cdot \frac{1}{9} \cdot \int \frac{3 u + 1}{u ^{2} - 1} d u

Expand

\frac{3 u + 1}{u ^{2} - 1} : \frac{3 u}{u ^{2} - 1} + \frac{1}{u ^{2} - 1}

= 2 \cdot \frac{1}{9} \cdot \int \frac{3 u}{u ^{2} - 1} + \frac{1}{u ^{2} - 1} d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 2 \cdot \frac{1}{9} (\int \frac{3 u}{u ^{2} - 1} d u + \int \frac{1}{u ^{2} - 1} d u)

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

\int \frac{1}{u ^{2} - 1} d u = ln u^{2} - 1 + u

= 2 \cdot \frac{1}{9} (3 u^{2} - 1 + ln u^{2} - 1 + u)

Substitute back

u = x - \frac{1}{3}

= 2 \cdot \frac{1}{9} 3 (x - \frac{1}{3})^{2} - 1 + ln (x - \frac{1}{3})^{2} - 1 + x - \frac{1}{3}

Simplify

2 \cdot \frac{1}{9} 3 (x - \frac{1}{3})^{2} - 1 + ln (x - \frac{1}{3})^{2} - 1 + x - \frac{1}{3} : \frac{2}{9} 3 (x - \frac{1}{3})^{2} - 1 + ln (x - \frac{1}{3})^{2} - 1 + x - \frac{1}{3}

= \frac{2}{9} 3 (x - \frac{1}{3})^{2} - 1 + ln (x - \frac{1}{3})^{2} - 1 + x - \frac{1}{3}

Add a constant to the solution

= \frac{2}{9} 3 (x - \frac{1}{3})^{2} - 1 + ln (x - \frac{1}{3})^{2} - 1 + x - \frac{1}{3} + C

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What is the integral of (2x)/(sqrt(9x^2-6x-8)) ?
The integral of (2x)/(sqrt(9x^2-6x-8)) is 2/9 (3sqrt((x-1/3)^2-1)+ln|sqrt((x-1/3)^2-1)+x-1/3 |)+C