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

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

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

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

- - x^{2} + 5 + C

Solution steps

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

Apply u-substitution

= \int - \frac{1}{2 u} d u

Take the constant out:

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

= - \frac{1}{2} \cdot \int \frac{1}{u} d u

Apply the Power Rule

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

Substitute back

u = - x^{2} + 5

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

Simplify

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

= - - x^{2} + 5

Add a constant to the solution

= - - x^{2} + 5 + C

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