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

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

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

Simplify

\frac{a + x}{a - x} : \frac{a + x}{a - x}

= \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

= \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

x \to - 10 lim (x^{2} - 11 x + 10)

y = 2 x - 8

x \to 1 lim (\frac{1}{x ( x + 2 )})

(1, - 7), (2, - 1)

m^{2} (\frac{c}{2} - \frac{m}{3})

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