What is the integral of 2/(x^2sqrt(9+x^2)) ?

The integral of 2/(x^2sqrt(9+x^2)) is -(2sqrt(9+x^2))/(9x)+C

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integral of 2/(x^2sqrt(9+x^2))

\int \frac{2}{x ^{2} 9 + x ^{2}} d x

- \frac{2 9 + x ^{2}}{9 x} + C

Solution steps

\int \frac{2}{x ^{2} 9 + x ^{2}} d x

Take the constant out:

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

= 2 \cdot \int \frac{1}{x ^{2} 9 + x ^{2}} d x

Apply Trigonometric Substitution

: \int \frac{sec ( u )}{9 tan ^{2} ( u )} d u

= 2 \cdot \int \frac{sec ( u )}{9 tan ^{2} ( u )} d u

Take the constant out:

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

= 2 \cdot \frac{1}{9} \cdot \int \frac{sec ( u )}{tan ^{2} ( u )} d u

Rewrite using trig identities

= 2 \cdot \frac{1}{9} \cdot \int \frac{cos ( u )}{sin ^{2} ( u )} d u

Apply u-substitution

: \int \frac{1}{v ^{2}} d v

= 2 \cdot \frac{1}{9} \cdot \int \frac{1}{v ^{2}} d v

Apply the Power Rule

: - \frac{1}{v}

= 2 \cdot \frac{1}{9} (- \frac{1}{v})

Substitute back

= 2 \cdot \frac{1}{9} (- \frac{1}{sin ( arctan ( \frac{1}{3} x ) )})

Simplify

2 \cdot \frac{1}{9} (- \frac{1}{sin ( arctan ( \frac{1}{3} x ) )}) : - \frac{2 9 + x ^{2}}{9 x}

= - \frac{2 9 + x ^{2}}{9 x}

Add a constant to the solution

= - \frac{2 9 + x ^{2}}{9 x} + C

x \to - 2 lim (\frac{x - 1}{3 x + 6})

(y^{4} - x^{2}) \cdot \frac{d y}{d x} = - x y

\int (x - 1)^{5} d x

e^{- t} cos (2 t)

What is the integral of 2/(x^2sqrt(9+x^2)) ?
The integral of 2/(x^2sqrt(9+x^2)) is -(2sqrt(9+x^2))/(9x)+C