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

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

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

integral

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

- 42 arctan (\frac{x}{2}) + 4 x + C

Solution steps

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

Apply Integral Substitution

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

Take the constant out:

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

= 42 \cdot \int \frac{u ^{2}}{u ^{2} + 1} d u

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

= 42 \cdot \int - \frac{1}{u ^{2} + 1} + 1 d u

Apply the Sum Rule:

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

= 42 (- \int \frac{1}{u ^{2} + 1} d u + \int 1 d u)

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

\int 1 d u = u

= 42 (- arctan (u) + u)

Substitute back

u = \frac{x}{2}

= 42 (- arctan (\frac{x}{2}) + \frac{x}{2})

Simplify

42 (- arctan (\frac{x}{2}) + \frac{x}{2}) : - 42 arctan (\frac{x}{2}) + 4 x

= - 42 arctan (\frac{x}{2}) + 4 x

Add a constant to the solution

= - 42 arctan (\frac{x}{2}) + 4 x + C

a^{2} - x^{2}

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

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

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

(e^{y})^{3}

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