What is the integral of x/(sqrt(7x^2+6)) ?

The integral of x/(sqrt(7x^2+6)) is 1/7 sqrt(7x^2+6)+C

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integral of x/(sqrt(7x^2+6))

\int \frac{x}{7 x ^{2} + 6} d x

\frac{1}{7} 7 x^{2} + 6 + C

Solution steps

\int \frac{x}{7 x ^{2} + 6} d x

Apply u-substitution

= \int \frac{1}{14 u} d u

Take the constant out:

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

= \frac{1}{14} \cdot \int \frac{1}{u} d u

Apply the Power Rule

= \frac{1}{14} \cdot 2 u^{\frac{1}{2}}

Substitute back

u = 7 x^{2} + 6

= \frac{1}{14} \cdot 2 (7 x^{2} + 6)^{\frac{1}{2}}

Simplify

\frac{1}{14} \cdot 2 (7 x^{2} + 6)^{\frac{1}{2}} : \frac{1}{7} 7 x^{2} + 6

= \frac{1}{7} 7 x^{2} + 6

Add a constant to the solution

= \frac{1}{7} 7 x^{2} + 6 + C

What is the integral of x/(sqrt(7x^2+6)) ?
The integral of x/(sqrt(7x^2+6)) is 1/7 sqrt(7x^2+6)+C