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

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

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

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

9 + x^{2} + C

Solution steps

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

Apply u-substitution

: \int \frac{1}{2 u} d u

= \int \frac{1}{2 u} d u

Take the constant out:

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

= \frac{1}{2} \cdot \int \frac{1}{u} d u

Apply the Power Rule

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

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

Substitute back

u = 9 + x^{2}

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

Simplify

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

= 9 + x^{2}

Add a constant to the solution

= 9 + x^{2} + C

\frac{d}{d x} (\frac{x ^{2} + 3}{x ^{2} + 1})

\frac{\partial}{\partial y} (\frac{x ^{2}}{4 - y ^{2}})

\frac{\partial}{\partial p} (\frac{p + q}{p + r})

\frac{1}{d + k e ^{d}}

\frac{d ^{2}}{d x ^{2}} y + 36 y = 0

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