What is the integral of 1/(x^2sqrt(4x^2+9)) ?

The integral of 1/(x^2sqrt(4x^2+9)) is -(sqrt(9+4x^2))/(9x)+C

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integral of 1/(x^2sqrt(4x^2+9))

\int \frac{1}{x ^{2} 4 x ^{2} + 9} d x

- \frac{9 + 4 x ^{2}}{9 x} + C

Solution steps

\int \frac{1}{x ^{2} 4 x ^{2} + 9} d x

Apply Trigonometric Substitution

= \int \frac{2 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

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

Rewrite using trig identities

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

Apply u-substitution

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

Apply the Power Rule

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

Substitute back

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

Simplify

\frac{2}{9} (- \frac{1}{sin ( arctan ( \frac{2}{3} x ) )}) : - \frac{9 + 4 x ^{2}}{9 x}

= - \frac{9 + 4 x ^{2}}{9 x}

Add a constant to the solution

= - \frac{9 + 4 x ^{2}}{9 x} + C

d er i v a t i v e f (x) = 6 x^{3} - \frac{27}{2} x^{2} + 3

d er i v a t i v e \frac{8}{7 x ^{2}}

d er i v a t i v e (5 x^{6} + 8 x^{3})^{4}

in v erse l a pl a ce \frac{2}{s ( 0.12 s + 1 )}

x y^{^{'}} = y + 7 x^{2} sin (x)

What is the integral of 1/(x^2sqrt(4x^2+9)) ?
The integral of 1/(x^2sqrt(4x^2+9)) is -(sqrt(9+4x^2))/(9x)+C