What is the integral of (2y)/(y^2-25) ?

The integral of (2y)/(y^2-25) is ln|y^2-25|+C

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integral of (2y)/(y^2-25)

\int \frac{2 y}{y ^{2} - 25} d y

ln y^{2} - 25 + C

Solution steps

\int \frac{2 y}{y ^{2} - 25} d y

Take the constant out:

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

= 2 \cdot \int \frac{y}{y ^{2} - 25} d y

Apply u-substitution

= 2 \cdot \int \frac{1}{2 u} 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} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 2 \cdot \frac{1}{2} ln ∣ u ∣

Substitute back

u = y^{2} - 25

= 2 \cdot \frac{1}{2} ln y^{2} - 25

Simplify

2 \cdot \frac{1}{2} ln y^{2} - 25 : ln y^{2} - 25

= ln y^{2} - 25

Add a constant to the solution

= ln y^{2} - 25 + C

What is the integral of (2y)/(y^2-25) ?
The integral of (2y)/(y^2-25) is ln|y^2-25|+C