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Rewrite the expression $\frac{1}{y^2-4}$ inside the integral in factored form

Rewrite the fraction $\frac{1}{\left(y+2\right)\left(y-2\right)}$ in $2$ simpler fractions using partial fraction decomposition

Find the values for the unknown coefficients: $A, B$. The first step is to multiply both sides of the equation from the previous step by $\left(y+2\right)\left(y-2\right)$

Multiplying polynomials

Simplifying

Assigning values to $y$ we obtain the following system of equations

Proceed to solve the system of linear equations

Rewrite as a coefficient matrix

Reducing the original matrix to a identity matrix using Gaussian Elimination

The integral of $\frac{1}{\left(y+2\right)\left(y-2\right)}$ in decomposed fraction equals

Expand the integral $\int\left(\frac{-1}{4\left(y+2\right)}+\frac{1}{4\left(y-2\right)}\right)dy$ into $2$ integrals using the sum rule for integrals, to then solve each integral separately

Take the constant $\frac{1}{4}$ out of the integral

Take the constant $\frac{1}{4}$ out of the integral

Apply the formula: $\int\frac{n}{x+b}dx$$=nsign\left(x\right)\ln\left(x+b\right)+C$, where $b=2$, $x=y$ and $n=-1$

Multiply the fraction and term in $\left(\frac{1}{4}\right)\left(-1\right)\ln\left|y+2\right|$

Apply the formula: $\int\frac{n}{x+b}dx$$=nsign\left(x\right)\ln\left(x+b\right)+C$, where $b=-2$, $x=y$ and $n=1$

Any expression multiplied by $1$ is equal to itself

The integral of a constant is equal to the constant times the integral's variable

As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$