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

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

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

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

2 arctan (x^{2}) + C

Solution steps

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

Take the constant out:

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

= 4 \cdot \int \frac{x}{x ^{4} + 1} d x

Apply u-substitution

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

= 4 \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

= 4 \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)

= 4 \cdot \frac{1}{2} arctan (u)

Substitute back

u = x^{2}

= 4 \cdot \frac{1}{2} arctan (x^{2})

Simplify

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

= 2 arctan (x^{2})

Add a constant to the solution

= 2 arctan (x^{2}) + C

\int 25 x^{2} d x

\frac{d}{d x} (c e^{- x})

2 t \frac{d y}{d t} + y = t^{3}

\frac{( x ^{2} - 8 x - 17 )}{( x + 2 )}

cos (e^{x^{4} c o s (x)})

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