To find the integral [tex]\(\int \frac{x^2+2}{x^2+3x+7} \, dx\)[/tex], follow these steps:
1. Simplify the Numerator: Notice that the numerator [tex]\(x^2 + 2\)[/tex] can be rewritten in a form that makes the integration more manageable. We do this by expressing [tex]\(x^2 + 2\)[/tex] in terms of the denominator [tex]\(x^2 + 3x + 7\)[/tex].
2. Rewrite the Numerator:
We observe that:
[tex]\[
x^2 + 3x + 7 = (x^2 + 3x) + 7
\][/tex]
To manipulate [tex]\(x^2 + 2\)[/tex], consider subtracting and adding the terms from the denominator:
[tex]\[
x^2 + 2 = (x^2 + 3x + 7) - 3x - 5
\][/tex]
Therefore:
[tex]\[
\frac{x^2 + 2}{x^2 + 3x + 7} = \frac{x^2 + 3x + 7 - 3x - 5}{x^2 + 3x + 7} = 1 - \frac{3x + 5}{x^2 + 3x + 7}
\][/tex]
3. Split the Integral: Now we can split the integral into two simpler integrals:
[tex]\[
\int \frac{x^2 + 2}{x^2 + 3x + 7} \, dx = \int 1 \, dx - \int \frac{3x + 5}{x^2 + 3x + 7} \, dx
\][/tex]
4. Integrate the First Term:
The integral of 1 with respect to [tex]\(x\)[/tex] is simply:
[tex]\[
\int 1 \, dx = x
\][/tex]
5. Integrate the Second Term:
To integrate [tex]\(\frac{3x + 5}{x^2 + 3x + 7}\)[/tex], we break it down further using partial fractions or appropriate substitutions.
a. Separate the Integral:
[tex]\[
\int \frac{3x + 5}{x^2 + 3x + 7} \, dx = \int \frac{3x + 3}{x^2 + 3x + 7} \, dx + \int \frac{2}{x^2 + 3x + 7} \, dx
\][/tex]
i. First Part: Notice [tex]\(3x + 3\)[/tex] in the numerator. We split this into:
[tex]\[
\int \frac{3x + 3}{x^2 + 3x + 7} \, dx = \int \frac{3(x + \frac{3}{2})}{x^2 + 3x + 7} \, dx
\][/tex]
Using substitution [tex]\(u = x^2 + 3x + 7\)[/tex], which gives us [tex]\(du = (2x + 3) dx\)[/tex]. Adjust the constants properly.
ii. Second Part: For the term [tex]\( \frac{2}{x^2 + 3x + 7} \)[/tex], complete the square:
[tex]\[
x^2 + 3x + 7 = (x + \frac{3}{2})^2 + \frac{19}{4}
\][/tex]
Then use a trigonometric substitution [tex]\(x + \frac{3}{2} = u\tan(\theta)\)[/tex].
6. Combine the Results:
Gather all parts:
[tex]\[
x - \frac{3}{2}\ln(x^2 + 3x + 7) - \frac{\sqrt{19}}{19} \arctan \left( \frac{2\sqrt{19}x + 3\sqrt{19}}{19} \right)
\][/tex]
Thus the final integral evaluation is:
[tex]\[
\boxed{x - \frac{3}{2} \ln(x^2 + 3x + 7) - \frac{\sqrt{19}}{19} \arctan \left( \frac{2 \sqrt{19}x}{19} + \frac{3\sqrt{19}}{19} \right) + C}
\][/tex]
where [tex]\(C\)[/tex] is the constant of integration.
