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

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

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

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

- \frac{1}{2 2} (ln \frac{2 - x ^{2}}{2} + 1 - ln \frac{2 - x ^{2}}{2} - 1) + C

Solution steps

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

Apply u-substitution

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

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

Take the constant out:

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

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

Apply Integral Substitution

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

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}{- v ^{2} + 1} d v

Use the common integral:

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

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

Substitute back

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

Simplify

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

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

Add a constant to the solution

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

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

2 x e^{2 x}

n = 9 \sum \infty e^{7 - 4 n}

\frac{\partial}{\partial x} (e^{7 x + 7 y})

x \to - 5 lim (\frac{x + 5}{5 - ∣ x ∣})

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