To evaluate the integral of the function

$\int \frac{7(x^2 + 5)}{x(x^2 + 7)} \, dx,$

we can first attempt to simplify the expression or decompose it into simpler terms using algebraic manipulation (such as partial fraction decomposition).

Step 1: Simplify the expression

We can rewrite the integrand:

$\frac{7(x^2 + 5)}{x(x^2 + 7)} = \frac{7x^2 + 35}{x(x^2 + 7)}.$

Now split this into two separate terms:

$\frac{7x^2 + 35}{x(x^2 + 7)} = \frac{7x^2}{x(x^2 + 7)} + \frac{35}{x(x^2 + 7)}.$

Simplifying each term:

$= \frac{7}{x^2 + 7} + \frac{35}{x(x^2 + 7)}.$

Step 2: Integrate each term separately

Now, let's evaluate the integrals of each term.

First term: $\int \frac{7}{x^2 + 7} \, dx$

This is a standard arctangent form:

$\int \frac{1}{x^2 + a^2} \, dx = \frac{1}{a} \arctan\left(\frac{x}{a}\right).$

So, we have:

$\int \frac{7}{x^2 + 7} \, dx = \frac{7}{\sqrt{7}} \arctan\left(\frac{x}{\sqrt{7}}\right).$

Second term: $\int \frac{35}{x(x^2 + 7)} \, dx$

We use partial fraction decomposition for this term. Let’s write:

$\frac{35}{x(x^2 + 7)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 7}.$

Multiply both sides by $x(x^2 + 7)$ to get:

$35 = A(x^2 + 7) + (Bx + C)x.$

Expanding:

$35 = A x^2 + 7A + Bx^2 + Cx.$

Now, equate the coefficients from both sides:

$A + B = 0 \quad \text{(coefficient of } x^2 \text{)}$ $C = 0 \quad \text{(coefficient of } x \text{)}$ $7A = 35 \quad \text{(constant term)}.$

From $7A = 35$, we get $A = 5$, and from $A + B = 0$, we get $B = -5$. Thus, we can rewrite the decomposition as:

$\frac{35}{x(x^2 + 7)} = \frac{5}{x} - \frac{5x}{x^2 + 7}.$

Now, integrate each term:

$\int \frac{5}{x} \, dx = 5 \ln|x|,$ $\int \frac{-5x}{x^2 + 7} \, dx = -\frac{5}{2} \ln(x^2 + 7).$

Step 3: Combine all terms

Now, combine all the integrals:

$\int \frac{7(x^2 + 5)}{x(x^2 + 7)} \, dx = \frac{7}{\sqrt{7}} \arctan\left(\frac{x}{\sqrt{7}}\right) + 5 \ln|x| - \frac{5}{2} \ln(x^2 + 7) + C.$

Final Answer:

$\int \frac{7(x^2 + 5)}{x(x^2 + 7)} \, dx = \frac{7}{\sqrt{7}} \arctan\left(\frac{x}{\sqrt{7}}\right) + 5 \ln|x| - \frac{5}{2} \ln(x^2 + 7) + C.$

Would you like any further explanation or have other questions about this? Here's a set of related questions for deeper understanding:

1. What is the process of partial fraction decomposition?
2. How do you identify the arctangent integral form?
3. Can you explain the general method for handling integrals involving $x(x^2 + a^2)$ in the denominator?
4. How do we integrate functions of the form $\frac{1}{x(x^2 + a^2)}$?
5. What would happen if the denominator had different powers of $x$?

Tip: Always try to simplify the integrand into easier terms (like partial fractions or simple forms) before performing the integration!