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

The integral of sqrt((a+x)/(a-x)) is 2aarctan(sqrt((a+x)/(a-x)))-sqrt(a+x)sqrt(a-x)+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

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

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

2 a arctan (\frac{a + x}{a - x}) - a + x a - x + C

Solution steps

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

Apply u-substitution

= \int \frac{4 a u ^{2}}{u ^{2} + 1 ( u ^{2} + 1 ) ^{\frac{3}{2}}} d u

Take the constant out:

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

= 4 a \cdot \int \frac{u ^{2}}{u ^{2} + 1 ( u ^{2} + 1 ) ^{\frac{3}{2}}} d u

(u^{2} + 1)^{\frac{3}{2}} = (u^{2} + 1) u^{2} + 1

= 4 a \cdot \int \frac{u ^{2}}{u ^{2} + 1 ( u ^{2} + 1 ) u ^{2} + 1} d u

u^{2} + 1 (u^{2} + 1) u^{2} + 1 = (u^{2} + 1)^{2}

= 4 a \cdot \int \frac{u ^{2}}{( u ^{2} + 1 ) ^{2}} d u

Take the partial fraction of

\frac{u ^{2}}{( u ^{2} + 1 ) ^{2}} : \frac{1}{u ^{2} + 1} - \frac{1}{( u ^{2} + 1 ) ^{2}}

= 4 a \cdot \int \frac{1}{u ^{2} + 1} - \frac{1}{( u ^{2} + 1 ) ^{2}} 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

= 4 a (\int \frac{1}{u ^{2} + 1} d u - \int \frac{1}{( u ^{2} + 1 ) ^{2}} d u)

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

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

= 4 a (arctan (u) - \frac{1}{2} (arctan (u) + \frac{u}{1 + u ^{2}}))

Substitute back

u = \frac{a + x}{a - x}

= 4 a arctan (\frac{a + x}{a - x}) - \frac{1}{2} arctan (\frac{a + x}{a - x}) + \frac{\frac{a + x}{a - x}}{1 + ( \frac{a + x}{a - x} ) ^{2}}

Simplify

4 a arctan (\frac{a + x}{a - x}) - \frac{1}{2} arctan (\frac{a + x}{a - x}) + \frac{\frac{a + x}{a - x}}{1 + ( \frac{a + x}{a - x} ) ^{2}} : 2 a arctan (\frac{a + x}{a - x}) - a + x a - x

= 2 a arctan (\frac{a + x}{a - x}) - a + x a - x

Add a constant to the solution

= 2 a arctan (\frac{a + x}{a - x}) - a + x a - x + C

What is the integral of sqrt((a+x)/(a-x)) ?
The integral of sqrt((a+x)/(a-x)) is 2aarctan(sqrt((a+x)/(a-x)))-sqrt(a+x)sqrt(a-x)+C