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

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

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

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

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

Solution steps

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

Long division

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

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

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

\int x^{2} d x = \frac{x ^{3}}{3}

\int 1 d x = x

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

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

Add a constant to the solution

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

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