What is the integral of 2/(4x^2+3) ?

The integral of 2/(4x^2+3) is 1/(sqrt(3))arctan(2/(sqrt(3))x)+C

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integral of 2/(4x^2+3)

\int \frac{2}{4 x ^{2} + 3} d x

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

Solution steps

\int \frac{2}{4 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}{4 x ^{2} + 3} d x

Apply Integral Substitution

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

Take the constant out:

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

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

Use the common integral:

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

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

Substitute back

u = \frac{2}{3} x

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

Simplify

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

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

Add a constant to the solution

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

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What is the integral of 2/(4x^2+3) ?
The integral of 2/(4x^2+3) is 1/(sqrt(3))arctan(2/(sqrt(3))x)+C