What is the integral of sqrt((3+x)/(1-x)) ?

The integral of sqrt((3+x)/(1-x)) is -sqrt(x+3)sqrt(-x+1)+4arctan(sqrt((x+3)/(-x+1)))+C

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

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

- x + 3 - x + 1 + 4 arctan (\frac{x + 3}{- x + 1}) + C

Solution steps

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

Apply u-substitution

: \int \frac{u}{- u + 4} d u

= \int \frac{u}{- u + 4} d u

Apply u-substitution

: \int - \frac{- v + 4}{v} d v

= \int - \frac{- v + 4}{v} d v

Take the constant out:

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

= - \int \frac{- v + 4}{v} d v

Expand

\frac{- v + 4}{v} : - 1 + \frac{4}{v}

= - \int - 1 + \frac{4}{v} d v

Apply u-substitution

: \int - \frac{4 w - 1}{w ^{2}} d w

= - \int - \frac{4 w - 1}{w ^{2}} d w

Take the constant out:

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

= - (- 4 \cdot \int \frac{w - 1}{w ^{2}} d w)

Apply Integration By Parts

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

= - (- 4 (- \frac{w - 1}{w} - \int - \frac{1}{2 w w - 1} d w))

Take the constant out:

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

= - (- 4 (- \frac{w - 1}{w} - (- \frac{1}{2} \cdot \int \frac{1}{w w - 1} d w)))

Apply u-substitution

: \int \frac{2}{t ^{2} + 1} d t

= - (- 4 (- \frac{w - 1}{w} - (- \frac{1}{2} \cdot \int \frac{2}{t ^{2} + 1} d t)))

Take the constant out:

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

= - (- 4 (- \frac{w - 1}{w} - (- \frac{1}{2} \cdot 2 \cdot \int \frac{1}{t ^{2} + 1} d t)))

Use the common integral:

\int \frac{1}{t ^{2} + 1} d t = arctan (t)

= - (- 4 (- \frac{w - 1}{w} - (- \frac{1}{2} \cdot 2 arctan (t))))

Substitute back

= - - 4 - \frac{\frac{4}{- ( 3 + x ) + 4} - 1}{\frac{4}{- ( 3 + x ) + 4}} - (- \frac{1}{2} \cdot 2 arctan (\frac{4}{- ( 3 + x ) + 4} - 1))

Simplify

- - 4 - \frac{\frac{4}{- ( 3 + x ) + 4} - 1}{\frac{4}{- ( 3 + x ) + 4}} - (- \frac{1}{2} \cdot 2 arctan (\frac{4}{- ( 3 + x ) + 4} - 1)) : - x + 3 - x + 1 + 4 arctan (\frac{x + 3}{- x + 1})

= - x + 3 - x + 1 + 4 arctan (\frac{x + 3}{- x + 1})

Add a constant to the solution

= - x + 3 - x + 1 + 4 arctan (\frac{x + 3}{- x + 1}) + C

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What is the integral of sqrt((3+x)/(1-x)) ?
The integral of sqrt((3+x)/(1-x)) is -sqrt(x+3)sqrt(-x+1)+4arctan(sqrt((x+3)/(-x+1)))+C