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

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

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integral of x/(sqrt(1+5x^2))

\int \frac{x}{1 + 5 x ^{2}} d x

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

Solution steps

\int \frac{x}{1 + 5 x ^{2}} d x

Apply u-substitution

= \int \frac{1}{10 u} d u

Take the constant out:

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

= \frac{1}{10} \cdot \int \frac{1}{u} d u

Apply the Power Rule

= \frac{1}{10} \cdot 2 u^{\frac{1}{2}}

Substitute back

u = 1 + 5 x^{2}

= \frac{1}{10} \cdot 2 (1 + 5 x^{2})^{\frac{1}{2}}

Simplify

\frac{1}{10} \cdot 2 (1 + 5 x^{2})^{\frac{1}{2}} : \frac{1}{5} 1 + 5 x^{2}

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

Add a constant to the solution

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

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