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

The integral of x/((1+x^2)^3) is -1/(4(1+x^2)^2)+C

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

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

- \frac{1}{4 ( 1 + x ^{2} ) ^{2}} + C

Solution steps

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

Apply u-substitution

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

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

Apply the Power Rule

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

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

Substitute back

u = 1 + x^{2}

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

Simplify

\frac{1}{2} (- \frac{1}{2 ( 1 + x ^{2} ) ^{2}}) : - \frac{1}{4 ( 1 + x ^{2} ) ^{2}}

= - \frac{1}{4 ( 1 + x ^{2} ) ^{2}}

Add a constant to the solution

= - \frac{1}{4 ( 1 + x ^{2} ) ^{2}} + C

(2 x + 5)^{2}

\int_{- 1}^{0} x^{\frac{1}{3}} - x^{\frac{2}{3}} d x

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

\int x^{2} x^{3} + 13 d x

2 x^{2} + 10 x + 9

What is the integral of x/((1+x^2)^3) ?
The integral of x/((1+x^2)^3) is -1/(4(1+x^2)^2)+C