What is the integral of 1/(xsqrt(12+x^2)) ?

The integral of 1/(xsqrt(12+x^2)) is 1/(2sqrt(3))ln|tan((arctan(1/(2sqrt(3))x))/2)|+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(12+x^2))

\int \frac{1}{x 12 + x ^{2}} d x

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

Solution steps

\int \frac{1}{x 12 + x ^{2}} d x

Apply Trigonometric Substitution

: \int \frac{sec ( u )}{2 3 tan ( u )} d u

= \int \frac{sec ( u )}{2 3 tan ( u )} d u

Take the constant out:

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

= \frac{1}{2 3} \cdot \int \frac{sec ( u )}{tan ( u )} d u

Apply u-substitution

: \int \frac{1}{v} d v

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

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= \frac{1}{2 3} ln ∣ v ∣

Substitute back

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

Add a constant to the solution

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

\int + \frac{y + 3}{( 3 - y ) ^{\frac{2}{3}}} d y

(4 + x) \frac{d y}{d x} = 9 y

\int - 7 e^{- x} d x

\int \frac{1}{x ^{2} + 2 x + 50} d x

x \to 1 + lim (\frac{2 + x}{1 - x})

What is the integral of 1/(xsqrt(12+x^2)) ?
The integral of 1/(xsqrt(12+x^2)) is 1/(2sqrt(3))ln|tan((arctan(1/(2sqrt(3))x))/2)|+C