What is the integral of x/(sqrt(4x^2+5)) ?

The integral of x/(sqrt(4x^2+5)) is 1/4 sqrt(4x^2+5)+C

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integral of x/(sqrt(4x^2+5))

\int \frac{x}{4 x ^{2} + 5} d x

\frac{1}{4} 4 x^{2} + 5 + C

Solution steps

\int \frac{x}{4 x ^{2} + 5} d x

Apply u-substitution

: \int \frac{1}{8 u} d u

= \int \frac{1}{8 u} d u

Take the constant out:

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

= \frac{1}{8} \cdot \int \frac{1}{u} d u

Apply the Power Rule

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

= \frac{1}{8} \cdot 2 u^{\frac{1}{2}}

Substitute back

u = 4 x^{2} + 5

= \frac{1}{8} \cdot 2 (4 x^{2} + 5)^{\frac{1}{2}}

Simplify

\frac{1}{8} \cdot 2 (4 x^{2} + 5)^{\frac{1}{2}} : \frac{1}{4} 4 x^{2} + 5

= \frac{1}{4} 4 x^{2} + 5

Add a constant to the solution

= \frac{1}{4} 4 x^{2} + 5 + C

\frac{\partial}{\partial x} (\frac{r x ( K ( r , x ) - x )}{K ( r , x ) + a x})

f (x) = 3 x^{2} + 1

y^{^{'}} = \frac{( 4 t + y )}{4 y - t}

5 3 x^{3} + x

\frac{e ^{\frac{1}{x}}}{x ^{3}}

What is the integral of x/(sqrt(4x^2+5)) ?
The integral of x/(sqrt(4x^2+5)) is 1/4 sqrt(4x^2+5)+C