What is the integral of 3/(5y^2+4) ?

The integral of 3/(5y^2+4) is 3/(2sqrt(5))arctan((sqrt(5))/2 y)+C

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integral of 3/(5y^2+4)

\int \frac{3}{5 y ^{2} + 4} d y

\frac{3}{2 5} arctan (\frac{5}{2} y) + C

Solution steps

\int \frac{3}{5 y ^{2} + 4} d y

Take the constant out:

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

= 3 \cdot \int \frac{1}{5 y ^{2} + 4} d y

Apply Integral Substitution

= 3 \cdot \int \frac{1}{2 5 ( 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 5} \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 5} arctan (u)

Substitute back

u = \frac{5}{2} y

= 3 \cdot \frac{1}{2 5} arctan (\frac{5}{2} y)

Simplify

3 \cdot \frac{1}{2 5} arctan (\frac{5}{2} y) : \frac{3}{2 5} arctan (\frac{5}{2} y)

= \frac{3}{2 5} arctan (\frac{5}{2} y)

Add a constant to the solution

= \frac{3}{2 5} arctan (\frac{5}{2} y) + C

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

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What is the integral of 3/(5y^2+4) ?
The integral of 3/(5y^2+4) is 3/(2sqrt(5))arctan((sqrt(5))/2 y)+C