(b) Find: $$\int \frac{x+3}{x^2+4x+5} dx$$ (integral, quadratic, antiderivative, calculus, integration)

Step 1: Analyze the Integrand

We need to find the integral of the given rational function. The denominator is a quadratic expression $x^2+4x+5$. Let's examine its derivative.

Step 2: Check Derivative of Denominator

The derivative of the denominator $x^2+4x+5$ is $2x+4$. We can manipulate the numerator to contain a multiple of this derivative.

Step 3: Manipulate the Numerator

We rewrite the numerator $x+3$ in terms of the derivative of the denominator, which is $2x+4$. We observe that $x+3 = \frac{1}{2}(2x+4) + 1$. This manipulation allows us to split the integral into two simpler parts.

Step 4: Split the Integral

Now we substitute the manipulated numerator back into the integral and split it into two separate integrals. The first integral will be a logarithmic form, and the second requires completing the square in the denominator.

Step 5: Evaluate the First Integral

For the first integral, we recognize the form $\int \frac{f'(x)}{f(x)} dx = \ln|f(x)| + C$. Here, $f(x) = x^2+4x+5$ and $f'(x) = 2x+4$. Therefore, the integral evaluates to $\frac{1}{2} \ln|x^2+4x+5|$.

Step 6: Complete the Square for the Second Integral

For the second integral, we complete the square in the denominator $x^2+4x+5$. We take half of the coefficient of $x$ (which is $4/2 = 2$), square it ($2^2=4$), and add and subtract it to the expression. This transforms $x^2+4x+5$ into $(x+2)^2 + 1$. This form is suitable for the arctangent integral formula.

Step 7: Evaluate the Second Integral

Using the formula $\int \frac{1}{a^2+x^2} dx = \frac{1}{a} \tan^{-1}(\frac{x}{a}) + C$, where for our integral $a=1$ and $x$ is replaced by $(x+2)$, the second integral evaluates to $\tan^{-1}(x+2)$.

Step 8: Combine the Results

Finally, we combine the results of the two integrals to get the complete solution. We replace $C_1+C_2$ with a single constant $C$ for the indefinite integral.

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