What is the integral of (2x)/(x^4+1) ?

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

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integral of (2x)/(x^4+1)

\int \frac{2 x}{x ^{4} + 1} d x

arctan (x^{2}) + C

Solution steps

\int \frac{2 x}{x ^{4} + 1} 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} + 1} d x

Apply u-substitution

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

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

Use the common integral:

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

= 2 \cdot \frac{1}{2} arctan (u)

Substitute back

u = x^{2}

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

Simplify

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

= arctan (x^{2})

Add a constant to the solution

= arctan (x^{2}) + C

f (x) = 12 x^{3} - \frac{1}{x}

x^{2} + 2 x + 2

(126 - 0.5 x) x - 50 x - 31.75

1 - \frac{d v}{d x} = cos (v)

\frac{d ^{2}}{d x ^{2}} (x^{2} + 9)

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