To find the indefinite integral [tex]\(\int \frac{x^2 + 10x + 2}{x^3 + 15x^2 + 6x + 7} \, dx\)[/tex], we need to examine the integrand and determine the best approach to solve it.
This integrand is a rational function, where the numerator and the denominator are polynomials. For such integrals, one effective method is to look for a simpler form that might make the integral easier to compute.
In this case, observe that the degree of the denominator (3) is higher than the degree of the numerator (2). This suggests that we might be able to apply a method involving specific simplifications such as partial fraction decomposition or recognizing it as a derivative of some logarithmic function.
1. Attempt to Recognize the Derivative of a Function:
Recognize that the derivative of the denominator might closely resemble the numerator in our function.
Let’s denote the denominator as follows:
[tex]\[
u(x) = x^3 + 15x^2 + 6x + 7
\][/tex]
2. Calculate the Derivative of [tex]\(u(x)\)[/tex]:
[tex]\[
u'(x) = \frac{d}{dx}(x^3 + 15x^2 + 6x + 7) = 3x^2 + 30x + 6
\][/tex]
3. Adjust the Given Numerator:
Notice that our numerator [tex]\(x^2 + 10x + 2\)[/tex] does not exactly match [tex]\(3x^2 + 30x + 6\)[/tex], but it is related.
Let's rewrite the original numerator in terms of [tex]\(u'(x)\)[/tex]:
[tex]\[
\frac{x^2 + 10x + 2}{x^3 + 15x^2 + 6x + 7}
\][/tex]
4. Factor Out a Constant to Match the Derivative:
By noticing this relationship, we can factor out a constant:
[tex]\[
\frac{1}{3} \cdot \frac{3x^2 + 30x + 6}{x^3 + 15x^2 + 6x + 7}
\][/tex]
So the integral becomes:
[tex]\[
\int \frac{x^2 + 10x + 2}{x^3 + 15x^2 + 6x + 7} \, dx = \frac{1}{3} \int \frac{3x^2 + 30x + 6}{x^3 + 15x^2 + 6x + 7} \, dx
\][/tex]
5. Recognize the Antiderivative:
Since [tex]\(\frac{3x^2 + 30x + 6}{x^3 + 15x^2 + 6x + 7} = \frac{u'(x)}{u(x)}\)[/tex], we have:
[tex]\[
\frac{1}{3} \int \frac{3x^2 + 30x + 6}{x^3 + 15x^2 + 6x + 7} \, dx = \frac{1}{3} \int \frac{u'(x)}{u(x)} \, dx
\][/tex]
6. Integrate:
The integral of [tex]\(\frac{u'(x)}{u(x)}\)[/tex] is [tex]\(\ln|u(x)|\)[/tex]:
[tex]\[
\frac{1}{3} \int \frac{u'(x)}{u(x)} \, dx = \frac{1}{3} \ln|u(x)| + C
\][/tex]
7. Substitute [tex]\(u(x)\)[/tex] Back:
[tex]\[
= \frac{1}{3} \ln|x^3 + 15x^2 + 6x + 7| + C
\][/tex]
Thus, the indefinite integral is:
[tex]\[
\int \frac{x^2 + 10x + 2}{x^3 + 15x^2 + 6x + 7} \, dx = \frac{1}{3} \ln|x^3 + 15x^2 + 6x + 7| + C
\][/tex]
