What is the integral of (x^2)/(2+x^6) ?

The integral of (x^2)/(2+x^6) is 1/(3sqrt(2))arctan((x^3)/(sqrt(2)))+C

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integral of (x^2)/(2+x^6)

\int \frac{x ^{2}}{2 + x ^{6}} d x

\frac{1}{3 2} arctan (\frac{x ^{3}}{2}) + C

Solution steps

\int \frac{x ^{2}}{2 + x ^{6}} d x

Apply u-substitution

= \int \frac{1}{3 ( 2 + u ^{2} )} d u

Take the constant out:

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

= \frac{1}{3} \cdot \int \frac{1}{2 + u ^{2}} d u

Apply Integral Substitution

= \frac{1}{3} \cdot \int \frac{1}{2 ( 1 + v ^{2} )} d v

Take the constant out:

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

= \frac{1}{3} \cdot \frac{1}{2} \cdot \int \frac{1}{1 + v ^{2}} d v

Use the common integral:

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

= \frac{1}{3} \cdot \frac{1}{2} arctan (v)

Substitute back

= \frac{1}{3} \cdot \frac{1}{2} arctan (\frac{x ^{3}}{2})

Simplify

\frac{1}{3} \cdot \frac{1}{2} arctan (\frac{x ^{3}}{2}) : \frac{1}{3 2} arctan (\frac{x ^{3}}{2})

= \frac{1}{3 2} arctan (\frac{x ^{3}}{2})

Add a constant to the solution

= \frac{1}{3 2} arctan (\frac{x ^{3}}{2}) + C

What is the integral of (x^2)/(2+x^6) ?
The integral of (x^2)/(2+x^6) is 1/(3sqrt(2))arctan((x^3)/(sqrt(2)))+C