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

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

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

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

\frac{1}{2} ln 2 x + 1 + 1 - \frac{1}{2} ln 2 x + 1 - 1 - 2 x + 1 - \frac{1}{2} ln ∣ x ∣ + C

Solution steps

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

Factor

1 - 2 x + 1 : - (- 1 + 2 x + 1)

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

Take the constant out:

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

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

Multiply by the conjugate of

- 1 + 2 x + 1 : \frac{2 x}{2 x + 1 + 1}

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

Expand

\frac{1}{\frac{x}{2 x + 1 + 1}} : \frac{2 x + 1}{x} + \frac{1}{x}

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

Apply the Sum Rule:

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

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

\int \frac{2 x + 1}{x} d x = - ln 2 x + 1 + 1 + ln 2 x + 1 - 1 + 2 2 x + 1

\int \frac{1}{x} d x = ln ∣ x ∣

= - \frac{1}{2} (- ln 2 x + 1 + 1 + ln 2 x + 1 - 1 + 2 2 x + 1 + ln ∣ x ∣)

Simplify

- \frac{1}{2} (- ln 2 x + 1 + 1 + ln 2 x + 1 - 1 + 2 2 x + 1 + ln ∣ x ∣) : \frac{1}{2} ln 2 x + 1 + 1 - \frac{1}{2} ln 2 x + 1 - 1 - 2 x + 1 - \frac{1}{2} ln ∣ x ∣

= \frac{1}{2} ln 2 x + 1 + 1 - \frac{1}{2} ln 2 x + 1 - 1 - 2 x + 1 - \frac{1}{2} ln ∣ x ∣

Add a constant to the solution

= \frac{1}{2} ln 2 x + 1 + 1 - \frac{1}{2} ln 2 x + 1 - 1 - 2 x + 1 - \frac{1}{2} ln ∣ x ∣ + C

\int \frac{4}{x ( ln ( x ) ) ^{4}} d x

\int 1 + y^{4} d y

\int \frac{( ln ( x ) ) ^{3}}{x ^{2}} d x

\int t^{2} e^{3 t} d t

\int \frac{e ^{3 x}}{e ^{2 x} + 1} d x

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