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

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

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

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

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

Solution steps

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

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

Simplify

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

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

Add a constant to the solution

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

x \to 0 lim (\frac{1 - cot ( x )}{\frac{1}{x}})

5 + x^{2}

\frac{\partial}{\partial x} (2 y - x^{2})

\frac{d}{d x} (\frac{( x ^{2} - 5 x )}{x + 1})

\int \frac{x + 5}{5 - x} d x

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