What is the integral of x/((x^2+1)^5) ?

The integral of x/((x^2+1)^5) is -1/(8(x^2+1)^4)+C

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integral of x/((x^2+1)^5)

\int \frac{x}{( x ^{2} + 1 ) ^{5}} d x

- \frac{1}{8 ( x ^{2} + 1 ) ^{4}} + C

Solution steps

\int \frac{x}{( x ^{2} + 1 ) ^{5}} d x

Apply u-substitution

: \int \frac{1}{2 u ^{5}} d u

= \int \frac{1}{2 u ^{5}} 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 ^{5}} d u

Apply the Power Rule

: - \frac{1}{4 u ^{4}}

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

Substitute back

u = x^{2} + 1

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

Simplify

\frac{1}{2} (- \frac{1}{4 ( x ^{2} + 1 ) ^{4}}) : - \frac{1}{8 ( x ^{2} + 1 ) ^{4}}

= - \frac{1}{8 ( x ^{2} + 1 ) ^{4}}

Add a constant to the solution

= - \frac{1}{8 ( x ^{2} + 1 ) ^{4}} + C

\int 2 x + y cos (x y) d x

\int \frac{1}{12 - 9 x ^{2}} d x

\int 5 x^{4} - \frac{4}{x ^{4}} d x

What is the integral of x/((x^2+1)^5) ?
The integral of x/((x^2+1)^5) is -1/(8(x^2+1)^4)+C