What is the integral of 1/(25+9y^2) ?

The integral of 1/(25+9y^2) is 1/15 arctan(3/5 y)+C

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integral of 1/(25+9y^2)

\int \frac{1}{25 + 9 y ^{2}} d y

\frac{1}{15} arctan (\frac{3}{5} y) + C

Solution steps

\int \frac{1}{25 + 9 y ^{2}} d y

Apply Integral Substitution

= \int \frac{1}{15 ( u ^{2} + 1 )} d u

Take the constant out:

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

= \frac{1}{15} \cdot \int \frac{1}{u ^{2} + 1} d u

Use the common integral:

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

= \frac{1}{15} arctan (u)

Substitute back

u = \frac{3}{5} y

= \frac{1}{15} arctan (\frac{3}{5} y)

Add a constant to the solution

= \frac{1}{15} arctan (\frac{3}{5} y) + C

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What is the integral of 1/(25+9y^2) ?
The integral of 1/(25+9y^2) is 1/15 arctan(3/5 y)+C