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

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

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

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

2 arctan (2 x - 1) + C

Solution steps

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

Apply u-substitution

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

= \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

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

Use the common integral:

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

= 2 arctan (u)

Substitute back

u = 2 x - 1

= 2 arctan (2 x - 1)

Add a constant to the solution

= 2 arctan (2 x - 1) + C

y = x (x - 10)

y = x^{4} + 2 e^{x} (0.2)

\int_{0}^{1} 1 - x^{2} d x

\frac{d}{d x} (cot^{2} (7 x))

\int \frac{k}{k - x} d x

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