What is the integral of 1/(x(2sqrt(x-1))) ?

The integral of 1/(x(2sqrt(x-1))) is arctan(sqrt(x-1))+C

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integral of 1/(x(2sqrt(x-1)))

\int \frac{1}{x ( 2 x - 1 )} d x

arctan (x - 1) + C

Solution steps

\int \frac{1}{x \cdot 2 x - 1} d x

Take the constant out:

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

= \frac{1}{2} \cdot \int \frac{1}{x x - 1} d x

Apply u-substitution

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

Take the constant out:

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

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

Use the common integral:

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

= \frac{1}{2} \cdot 2 arctan (u)

Substitute back

u = x - 1

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

Simplify

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

= arctan (x - 1)

Add a constant to the solution

= arctan (x - 1) + C

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