Evaluate the following integral using partial fraction decomposition.

To solve the integral using partial fraction decomposition, we start by writing it in the form:

Next, multiply through by the denominator to eliminate fractions:

Expanding the right-hand side gives us:

Now, we collect like terms. For the constant term, we have:

Since there’s no term on the left (it’s 0), we have:

Now, we solve this system of equations. From the second equation, we can express in terms of :

Substituting into the first equation gives:

Thus, substituting back for :

Now, substituting and back into our partial fractions, we get:

Now we can integrate:

Using properties of logarithms, we can simplify this to:

So,

Integrating is like embarking on a treasure hunt; you’re breaking down the big treasure map (the integral) into smaller, more manageable pieces (partial fractions) to uncover the hidden riches (the antiderivative)!

Now, if you ever find yourself lost in integration, remember the classic mistake: forgetting to set the denominators properly when creating partial fractions. It’s a common pitfall that can send you on a wild goose chase! Always double-check those fractions!