What is the integral of (5y)/(25y^2+9) ?

The integral of (5y)/(25y^2+9) is 1/10 ln|25y^2+9|+C

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integral of (5y)/(25y^2+9)

\int \frac{5 y}{25 y ^{2} + 9} d y

\frac{1}{10} ln 25 y^{2} + 9 + C

Solution steps

\int \frac{5 y}{25 y ^{2} + 9} d y

Take the constant out:

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

= 5 \cdot \int \frac{y}{25 y ^{2} + 9} d y

Apply u-substitution

: \int \frac{1}{50 u} d u

= 5 \cdot \int \frac{1}{50 u} d u

Take the constant out:

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

= 5 \cdot \frac{1}{50} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 5 \cdot \frac{1}{50} ln ∣ u ∣

Substitute back

u = 25 y^{2} + 9

= 5 \cdot \frac{1}{50} ln 25 y^{2} + 9

Simplify

5 \cdot \frac{1}{50} ln 25 y^{2} + 9 : \frac{1}{10} ln 25 y^{2} + 9

= \frac{1}{10} ln 25 y^{2} + 9

Add a constant to the solution

= \frac{1}{10} ln 25 y^{2} + 9 + C

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What is the integral of (5y)/(25y^2+9) ?
The integral of (5y)/(25y^2+9) is 1/10 ln|25y^2+9|+C