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

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

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integral of 3/(x^2+2)

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

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

Solution steps

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

Take the constant out:

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

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

Apply Integral Substitution

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

Take the constant out:

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

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

Use the common integral:

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

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

Substitute back

u = \frac{x}{2}

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

Simplify

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

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

Add a constant to the solution

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

\int 18 sec^{4} (x) d x

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\int 2 sin^{3} (2 x) cos^{3} (2 x) d x

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\frac{d y}{d x} = x^{6} y^{- 7}, y (0) = 4

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