What is the integral of 1/(xsqrt(2x-3)) ?

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

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integral of 1/(xsqrt(2x-3))

\int \frac{1}{x 2 x - 3} d x

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

Solution steps

\int \frac{1}{x 2 x - 3} d x

Apply u-substitution

: \int \frac{2}{u ^{2} + 3} d u

= \int \frac{2}{u ^{2} + 3} 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} + 3} d u

Apply Integral Substitution

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

Take the constant out:

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

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

Use the common integral:

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

= 2 \cdot \frac{1}{3} arctan (v)

Substitute back

= 2 \cdot \frac{1}{3} arctan (\frac{2 x - 3}{3})

Simplify

2 \cdot \frac{1}{3} arctan (\frac{2 x - 3}{3}) : \frac{2}{3} arctan (\frac{1}{3} (2 x - 3))

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

Add a constant to the solution

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

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