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

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

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

integral

\frac{4 x}{( x ^{2} + 1 )}

2 ln x^{2} + 1 + C

Solution steps

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

Apply u-substitution

: \int \frac{1}{2 u} d u

= 4 \cdot \int \frac{1}{2 u} 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} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 4 \cdot \frac{1}{2} ln ∣ u ∣

Substitute back

u = x^{2} + 1

= 4 \cdot \frac{1}{2} ln x^{2} + 1

Simplify

4 \cdot \frac{1}{2} ln x^{2} + 1 : 2 ln x^{2} + 1

= 2 ln x^{2} + 1

Add a constant to the solution

= 2 ln x^{2} + 1 + C

cosh^{2} (x)

1 - cot^{2} (x)

\frac{- 2}{x ^{3}}

\frac{ln ( x ^{2} )}{x ^{3}}

\frac{( x + 2 )}{( x ^{2} + x + 1 )}

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