What is the integral of (2x)/(x^4+9) ?

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

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integral of (2x)/(x^4+9)

\int \frac{2 x}{x ^{4} + 9} d x

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

Solution steps

\int \frac{2 x}{x ^{4} + 9} d x

Take the constant out:

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

= 2 \cdot \int \frac{x}{x ^{4} + 9} d x

Apply u-substitution

: \int \frac{1}{2 ( u ^{2} + 9 )} d u

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

Take the constant out:

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

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

Apply Integral Substitution

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

Take the constant out:

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

= 2 \cdot \frac{1}{2} \cdot \frac{1}{3} \cdot \int \frac{1}{v ^{2} + 1} d v

Use the common integral:

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

= 2 \cdot \frac{1}{2} \cdot \frac{1}{3} arctan (v)

Substitute back

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

Simplify

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

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

Add a constant to the solution

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

\int (\frac{1}{x ( 1 + x )}) d x

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