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

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

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

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

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

Solution steps

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

Take the constant out:

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

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

\frac{1}{3 x ^{2} + 3} = \frac{1}{3} \frac{1}{x ^{2} + 1}

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

Take the constant out:

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

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

Use the common integral:

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

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

Simplify

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

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

Add a constant to the solution

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

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