What is the integral of x/(sqrt((1+x^2)^3)) ?

The integral of x/(sqrt((1+x^2)^3)) is -1/(sqrt(1+x^2))+C

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integral of x/(sqrt((1+x^2)^3))

\int \frac{x}{( 1 + x ^{2} ) ^{3}} d x

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

Solution steps

\int \frac{x}{( 1 + x ^{2} ) ^{3}} d x

(1 + x^{2})^{3} = (1 + x^{2})^{\frac{3}{2}},

assuming

(1 + x^{2}) \geq 0

= \int \frac{x}{( 1 + x ^{2} ) ^{\frac{3}{2}}} d x

Apply u-substitution

= \int \frac{1}{2 u ^{\frac{3}{2}}} 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 ^{\frac{3}{2}}} d u

Apply the Power Rule

= \frac{1}{2} (- \frac{2}{u})

Substitute back

u = 1 + x^{2}

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

Simplify

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

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

Add a constant to the solution

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

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What is the integral of x/(sqrt((1+x^2)^3)) ?
The integral of x/(sqrt((1+x^2)^3)) is -1/(sqrt(1+x^2))+C