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

The integral of 1/(x+sqrt(x-1)) is ln|4x+4sqrt(x-1)|-2/(sqrt(3))arctan((2sqrt(x-1)+1)/(sqrt(3)))+C

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

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

ln 4 x + 4 x - 1 - \frac{2}{3} arctan (\frac{2 x - 1 + 1}{3}) + C

Solution steps

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

Apply u-substitution

: \int \frac{1}{u + 1 + u} d u

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

Apply u-substitution

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

= \int \frac{2 v}{v ^{2} + 1 + v} d v

Take the constant out:

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

= 2 \cdot \int \frac{v}{v ^{2} + 1 + v} d v

Complete the square

v^{2} + 1 + v : (v + \frac{1}{2})^{2} + \frac{3}{4}

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

Apply u-substitution

: \int \frac{2 ( 2 w - 1 )}{4 w ^{2} + 3} d w

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

Take the constant out:

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

= 2 \cdot 2 \cdot \int \frac{2 w - 1}{4 w ^{2} + 3} d w

Expand

\frac{2 w - 1}{4 w ^{2} + 3} : \frac{2 w}{4 w ^{2} + 3} - \frac{1}{4 w ^{2} + 3}

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

\int \frac{2 w}{4 w ^{2} + 3} d w = \frac{1}{4} ln 4 w^{2} + 3

\int \frac{1}{4 w ^{2} + 3} d w = \frac{1}{2 3} arctan (\frac{2}{3} w)

= 2 \cdot 2 (\frac{1}{4} ln 4 w^{2} + 3 - \frac{1}{2 3} arctan (\frac{2}{3} w))

Substitute back

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

Simplify

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

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

Add a constant to the solution

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

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What is the integral of 1/(x+sqrt(x-1)) ?
The integral of 1/(x+sqrt(x-1)) is ln|4x+4sqrt(x-1)|-2/(sqrt(3))arctan((2sqrt(x-1)+1)/(sqrt(3)))+C