What is the integral of x/(5+x^4) ?

The integral of x/(5+x^4) is 1/(2sqrt(5))arctan((x^2)/(sqrt(5)))+C

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integral of x/(5+x^4)

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

\frac{1}{2 5} arctan (\frac{x ^{2}}{5}) + C

Solution steps

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

Apply u-substitution

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

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

Apply Integral Substitution

= \frac{1}{2} \cdot \int \frac{1}{5 ( 1 + v ^{2} )} d v

Take the constant out:

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

= \frac{1}{2} \cdot \frac{1}{5} \cdot \int \frac{1}{1 + v ^{2}} d v

Use the common integral:

\int \frac{1}{1 + v ^{2}} d v = arctan (v)

= \frac{1}{2} \cdot \frac{1}{5} arctan (v)

Substitute back

= \frac{1}{2} \cdot \frac{1}{5} arctan (\frac{x ^{2}}{5})

Simplify

\frac{1}{2} \cdot \frac{1}{5} arctan (\frac{x ^{2}}{5}) : \frac{1}{2 5} arctan (\frac{x ^{2}}{5})

= \frac{1}{2 5} arctan (\frac{x ^{2}}{5})

Add a constant to the solution

= \frac{1}{2 5} arctan (\frac{x ^{2}}{5}) + C

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