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

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

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

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

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

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

: \int \frac{1}{18 u} d u

= - \int \frac{1}{18 u} 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} d u

Apply the Power Rule

: 2 u^{\frac{1}{2}}

= - \frac{1}{18} \cdot 2 u^{\frac{1}{2}}

Substitute back

u = 9 x^{2} + 2

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

Simplify

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

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

Add a constant to the solution

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

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