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

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

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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

= - \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

: 2 u^{\frac{1}{2}}

= - \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

sin (x) + cos (x)

\int \frac{5 x - 2}{( x + 3 ) ^{2}} d x

\int \frac{1}{csc ( v )} d v

\int \frac{2 - z}{2 + z} d z

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