Question - Find the Integral of a Function

\( \int f(x) \,dx = \int \frac{3x-5}{x^2+2x+1} \,dx \)

Let's factor the denominator \( x^2 + 2x + 1 = (x+1)^2 \).

Now, split the fraction into two parts: \( \frac{3x-5}{(x+1)^2} = \frac{Ax+B}{x+1} + \frac{C}{(x+1)^2} \)

Find A, B, and C by multiplying both sides by \( (x+1)^2 \) and equating coefficients:

\( 3x - 5 = Ax + B(x+1) + C \)

Solve for A, B, and C, we get \( A = 3, B = -3, C = -2 \).

Now integrate each term separately:

\( \int \frac{3x-3}{x+1} \,dx - \int \frac{2}{(x+1)^2} \,dx \)

For the first term, simplify to \( \int 3 - \frac{3}{x+1} \,dx \)

Integrate directly:

\( \int 3 \,dx - 3\int \frac{1}{x+1} \,dx - \int \frac{2}{(x+1)^2} \,dx \)

The integrals are:

\( 3x - 3\ln|x+1| - \frac{2}{x+1} + C \)

Where \( C \) is the constant of integration.