What is the integral of x/(9+25x^4) ?

The integral of x/(9+25x^4) is 1/30 arctan(5/3 x^2)+C

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integral of x/(9+25x^4)

\int \frac{x}{9 + 25 x ^{4}} d x

\frac{1}{30} arctan (\frac{5}{3} x^{2}) + C

Solution steps

\int \frac{x}{9 + 25 x ^{4}} d x

Apply u-substitution

= \int \frac{1}{2 ( 9 + 25 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}{9 + 25 u ^{2}} d u

Apply Integral Substitution

= \frac{1}{2} \cdot \int \frac{1}{15 ( v ^{2} + 1 )} 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}{15} \cdot \int \frac{1}{v ^{2} + 1} d v

Use the common integral:

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

= \frac{1}{2} \cdot \frac{1}{15} arctan (v)

Substitute back

= \frac{1}{2} \cdot \frac{1}{15} arctan (\frac{5}{3} x^{2})

Simplify

\frac{1}{2} \cdot \frac{1}{15} arctan (\frac{5}{3} x^{2}) : \frac{1}{30} arctan (\frac{5}{3} x^{2})

= \frac{1}{30} arctan (\frac{5}{3} x^{2})

Add a constant to the solution

= \frac{1}{30} arctan (\frac{5}{3} x^{2}) + C

What is the integral of x/(9+25x^4) ?
The integral of x/(9+25x^4) is 1/30 arctan(5/3 x^2)+C