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Popular >

integral of sqrt((3x+1)/(3x-1))

Solution

\int \frac{3 x + 1}{3 x - 1} d x

Solution

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

Solution steps

\int \frac{3 x + 1}{3 x - 1} d x

Apply u-substitution

: \int \frac{u + 2}{3 u} d u

= \int \frac{u + 2}{3 u} d u

Take the constant out:

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

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

Combine same powers :

\frac{x}{y} = \frac{x}{y}

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

\frac{u + 2}{u} = (\frac{u}{u} + \frac{2}{u})

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

\frac{u}{u} = 1

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

Apply u-substitution

: \int - \frac{2 v + 1}{v ^{2}} d v

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

Take the constant out:

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

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

Apply Integration By Parts

: - \frac{v + 1}{v} - \int - \frac{1}{2 v v + 1} d v

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

Take the constant out:

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

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

Apply u-substitution

: \int \frac{2}{w ^{2} - 1} d w

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

Take the constant out:

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

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

Factor

w^{2} - 1 : - (- w^{2} + 1)

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

Take the constant out:

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

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

Use the common integral:

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

= \frac{1}{3} (- 2 (- \frac{v + 1}{v} - (- \frac{1}{2} \cdot 2 (- (\frac{ln ∣ w + 1 ∣}{2} - \frac{ln ∣ w - 1 ∣}{2})))))

Substitute back

= \frac{1}{3} - 2 - \frac{\frac{2}{3 x - 1} + 1}{\frac{2}{3 x - 1}} - - \frac{1}{2} \cdot 2 - \frac{ln \frac{2}{3 x - 1} + 1 + 1}{2} - \frac{ln \frac{2}{3 x - 1} + 1 - 1}{2}

Simplify

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

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

Add a constant to the solution

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

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