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

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

= \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

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