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

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

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

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

- x^{2} + 1 + C

Solution steps

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

Take the constant out:

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

= - \int \frac{x}{x ^{2} + 1} 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} + 1

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

Simplify

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

= - x^{2} + 1

Add a constant to the solution

= - x^{2} + 1 + C

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