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

The integral of sqrt((3+x)/(3-x)) is -sqrt(x+3)sqrt(3-x)+6arctan(sqrt((x+3)/(3-x)))+C

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

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

- x + 3 3 - x + 6 arctan (\frac{x + 3}{3 - x}) + C

Solution steps

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

Apply u-substitution

: \int - \frac{- u + 6}{u} d u

= \int - \frac{- u + 6}{u} d u

Take the constant out:

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

= - \int \frac{- u + 6}{u} d u

Expand

\frac{- u + 6}{u} : - 1 + \frac{6}{u}

= - \int - 1 + \frac{6}{u} d u

Apply u-substitution

: \int - \frac{6 v - 1}{v ^{2}} d v

= - \int - \frac{6 v - 1}{v ^{2}} d v

Take the constant out:

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

= - (- 6 \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

= - (- 6 (- \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

= - (- 6 (- \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

= - (- 6 (- \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

= - (- 6 (- \frac{v - 1}{v} - (- \frac{1}{2} \cdot 2 \cdot \int \frac{1}{w ^{2} + 1} d w)))

Use the common integral:

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

= - (- 6 (- \frac{v - 1}{v} - (- \frac{1}{2} \cdot 2 arctan (w))))

Substitute back

= - - 6 - \frac{\frac{6}{3 - x} - 1}{\frac{6}{3 - x}} - (- \frac{1}{2} \cdot 2 arctan (\frac{6}{3 - x} - 1))

Simplify

- - 6 - \frac{\frac{6}{3 - x} - 1}{\frac{6}{3 - x}} - (- \frac{1}{2} \cdot 2 arctan (\frac{6}{3 - x} - 1)) : - x + 3 3 - x + 6 arctan (\frac{x + 3}{3 - x})

= - x + 3 3 - x + 6 arctan (\frac{x + 3}{3 - x})

Add a constant to the solution

= - x + 3 3 - x + 6 arctan (\frac{x + 3}{3 - x}) + C

x^{^{''}} + x^{^{'}} + 3 x = 0

\int \frac{8 x ^{3}}{x - 4} d x

\frac{x}{3 x + 5}

\frac{1}{∣ x ∣ + 1} + \frac{1}{∣ x - 2 ∣ + 1}

\frac{d}{d x} (\frac{4374}{x ^{2}})

What is the integral of sqrt((3+x)/(3-x)) ?
The integral of sqrt((3+x)/(3-x)) is -sqrt(x+3)sqrt(3-x)+6arctan(sqrt((x+3)/(3-x)))+C