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

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

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

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

- \frac{2 + x ^{2}}{2 x} + C

Solution steps

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

Apply Trigonometric Substitution

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

Rewrite using trig identities

= \frac{1}{2} \cdot \int \frac{cos ( u )}{sin ^{2} ( u )} d u

Apply u-substitution

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

Apply the Power Rule

= \frac{1}{2} (- \frac{1}{v})

Substitute back

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

Simplify

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

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

Add a constant to the solution

= - \frac{2 + x ^{2}}{2 x} + C

\int (e^{x} + 2 sec^{2} (x) + \frac{4}{x} + x) d x

\int cot (x) (2 + ln ∣ sin (x) ∣) d x

\int sec^{2} (θ) e^{3 t a n (θ)} d θ

\int 4 x^{3} + 3 x^{8} d x

\int \frac{1}{x} d ln (x) d x

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