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}{2 u 2 + u} d u

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

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}{u 2 + u} d u

Apply u-substitution

: \int \frac{2}{v ^{2} - 2} d v

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

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}{v ^{2} - 2} d v

Apply Integral Substitution

= \frac{1}{2} \cdot 2 \cdot \int \frac{1}{2 ( w ^{2} - 1 )} d w

Take the constant out:

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

= \frac{1}{2} \cdot 2 \cdot \frac{1}{2} \cdot \int \frac{1}{w ^{2} - 1} d w

Factor

w^{2} - 1 : - (- w^{2} + 1)

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

Take the constant out:

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

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

Use the common integral:

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

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

Substitute back

= \frac{1}{2} \cdot 2 \cdot \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} \cdot 2 \cdot \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

x = y^{2} - 1, x = 1 - y^{2}

12 x^{2} - 48 x

\frac{d y}{d x} = y + 3

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

\int sin (x) cos^{5} (x) d 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