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

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

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

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

- \frac{1}{x ^{2} + 1} + C

Solution steps

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

Factor

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

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

Apply u-substitution

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

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

Apply the Power Rule

: - \frac{1}{u}

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

Substitute back

u = x^{2} + 1

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

Simplify

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

= - \frac{1}{x ^{2} + 1}

Add a constant to the solution

= - \frac{1}{x ^{2} + 1} + C

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