What is the integral of (1/(3+2x^2)) ?

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

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integral of (1/(3+2x^2))

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

\frac{1}{6} arctan (\frac{2}{3} x) + C

Solution steps

\int \frac{1}{3 + 2 x ^{2}} d x

Apply Integral Substitution

= \int \frac{1}{6 ( 1 + u ^{2} )} d u

Take the constant out:

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

= \frac{1}{6} \cdot \int \frac{1}{1 + u ^{2}} d u

Use the common integral:

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

= \frac{1}{6} arctan (u)

Substitute back

u = \frac{2}{3} x

= \frac{1}{6} arctan (\frac{2}{3} x)

Combine same powers :

\frac{x}{y} = \frac{x}{y}

= \frac{1}{6} arctan (\frac{2}{3} x)

Add a constant to the solution

= \frac{1}{6} arctan (\frac{2}{3} x) + C

t \to 0 + lim (\frac{sin ( t )}{t})

\frac{d}{d x} (x^{3} (4 - x^{2}))

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

n = 0 \sum \infty \frac{e}{n}

\frac{\partial}{\partial x} (ln (t^{2} + 1))

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