What is the integral of x/(sqrt(4+9x^2)^3) ?

The integral of x/(sqrt(4+9x^2)^3) is -1/(9sqrt(9x^2+4))+C

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integral of x/(sqrt(4+9x^2)^3)

\int \frac{x}{4 + 9 x ^{2} ^{3}} d x

- \frac{1}{9 9 x ^{2} + 4} + C

Solution steps

\int \frac{x}{( 4 + 9 x ^{2} ) ^{3}} d x

(4 + 9 x^{2})^{3} = (4 + 9 x^{2})^{\frac{3}{2}}

= \int \frac{x}{( 4 + 9 x ^{2} ) ^{\frac{3}{2}}} d x

Apply u-substitution

: \int \frac{1}{18 u ^{\frac{3}{2}}} d u

= \int \frac{1}{18 u ^{\frac{3}{2}}} d u

Take the constant out:

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

= \frac{1}{18} \cdot \int \frac{1}{u ^{\frac{3}{2}}} d u

Apply the Power Rule

: - \frac{2}{u}

= \frac{1}{18} (- \frac{2}{u})

Substitute back

u = 4 + 9 x^{2}

= \frac{1}{18} (- \frac{2}{4 + 9 x ^{2}})

Simplify

\frac{1}{18} (- \frac{2}{4 + 9 x ^{2}}) : - \frac{1}{9 9 x ^{2} + 4}

= - \frac{1}{9 9 x ^{2} + 4}

Add a constant to the solution

= - \frac{1}{9 9 x ^{2} + 4} + C

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What is the integral of x/(sqrt(4+9x^2)^3) ?
The integral of x/(sqrt(4+9x^2)^3) is -1/(9sqrt(9x^2+4))+C