What is the integral of 8sqrt((1+x)/(1-x)) ?

The integral of 8sqrt((1+x)/(1-x)) is -8(sqrt((x+1)/(1-x))(1-x)+2arcsin(1/(sqrt(2))sqrt(1-x)))+C

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integral of 8sqrt((1+x)/(1-x))

\int 8 \frac{1 + x}{1 - x} d x

- 8 (\frac{x + 1}{1 - x} (1 - x) + 2 arcsin (\frac{1}{2} 1 - x)) + C

Solution steps

\int 8 \frac{1 + x}{1 - x} d x

Take the constant out:

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

= 8 \cdot \int \frac{1 + x}{1 - x} d x

Apply u-substitution

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

= 8 \cdot \int - \frac{- u + 2}{u} d u

Take the constant out:

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

= 8 (- \int \frac{- u + 2}{u} d u)

Apply Integration By Parts

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

= 8 (- (\frac{- u + 2}{u} u - \int - \frac{1}{u ^{\frac{1}{2}} - u + 2} d u))

Take the constant out:

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

= 8 (- (\frac{- u + 2}{u} u - (- \int \frac{1}{u ^{\frac{1}{2}} - u + 2} d u)))

Apply u-substitution

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

= 8 (- (\frac{- u + 2}{u} u - (- \int \frac{2}{- v ^{2} + 2} d v)))

Take the constant out:

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

= 8 (- (\frac{- u + 2}{u} u - (- 2 \cdot \int \frac{1}{- v ^{2} + 2} d v)))

Apply Trigonometric Substitution

: \int 1 d w

= 8 (- (\frac{- u + 2}{u} u - (- 2 \cdot \int 1 d w)))

Integral of a constant:

\int a d x = a x

= 8 (- (\frac{- u + 2}{u} u - (- 2 \cdot 1 \cdot w)))

Substitute back

= 8 (- (\frac{- ( 1 - x ) + 2}{1 - x} (1 - x) - (- 2 \cdot 1 \cdot arcsin (\frac{1}{2} (1 - x)^{\frac{1}{2}}))))

Simplify

8 (- (\frac{- ( 1 - x ) + 2}{1 - x} (1 - x) - (- 2 \cdot 1 \cdot arcsin (\frac{1}{2} (1 - x)^{\frac{1}{2}})))) : - 8 (\frac{x + 1}{1 - x} (1 - x) + 2 arcsin (\frac{1}{2} 1 - x))

= - 8 (\frac{x + 1}{1 - x} (1 - x) + 2 arcsin (\frac{1}{2} 1 - x))

Add a constant to the solution

= - 8 (\frac{x + 1}{1 - x} (1 - x) + 2 arcsin (\frac{1}{2} 1 - x)) + C

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What is the integral of 8sqrt((1+x)/(1-x)) ?
The integral of 8sqrt((1+x)/(1-x)) is -8(sqrt((x+1)/(1-x))(1-x)+2arcsin(1/(sqrt(2))sqrt(1-x)))+C