What is the integral of x/(sqrt(3x^2+5)) ?

The integral of x/(sqrt(3x^2+5)) is 1/3 sqrt(3x^2+5)+C

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integral of x/(sqrt(3x^2+5))

\int \frac{x}{3 x ^{2} + 5} d x

\frac{1}{3} 3 x^{2} + 5 + C

Solution steps

\int \frac{x}{3 x ^{2} + 5} d x

Apply u-substitution

: \int \frac{1}{6 u} d u

= \int \frac{1}{6 u} d u

Take the constant out:

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

= \frac{1}{6} \cdot \int \frac{1}{u} d u

Apply the Power Rule

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

= \frac{1}{6} \cdot 2 u^{\frac{1}{2}}

Substitute back

u = 3 x^{2} + 5

= \frac{1}{6} \cdot 2 (3 x^{2} + 5)^{\frac{1}{2}}

Simplify

\frac{1}{6} \cdot 2 (3 x^{2} + 5)^{\frac{1}{2}} : \frac{1}{3} 3 x^{2} + 5

= \frac{1}{3} 3 x^{2} + 5

Add a constant to the solution

= \frac{1}{3} 3 x^{2} + 5 + C

\frac{d}{d x} (\frac{4}{( x + 2 ) ^{2} + 16})

\frac{\partial}{\partial x} ((e^{x} - x) cos (y))

\frac{d}{d t} (- 32 t)

\int 25 - x^{2} - 3 d x

x \to 3 + lim (\frac{1}{x + 3})

What is the integral of x/(sqrt(3x^2+5)) ?
The integral of x/(sqrt(3x^2+5)) is 1/3 sqrt(3x^2+5)+C