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solve with partial fraction differentiation.

The integral provided is:

\[
\int \frac{x^4 + 1}{x^3 + 7x} \, dx
\]

### Step 1: Simplify the integrand
We start by factoring the denominator. Notice that:

\[
x^3 + 7x = x(x^2 + 7)
\]

Thus, the integral becomes:

\[
\int \frac{x^4 + 1}{x(x^2 + 7)} \, dx
\]

### Step 2: Polynomial long division
The degree of the numerator \(x^4 + 1\) is greater than the degree of the denominator \(x(x^2 + 7)\), so we need to perform polynomial long division first.

1. Divide \(x^4\) by \(x^3\), which gives \(x\).
2. Multiply \(x\) by \(x^3 + 7x\) to get \(x^4 + 7x^2\).
3. Subtract \(x^4 + 7x^2\) from \(x^4 + 1\):

\[
(x^4 + 1) - (x^4 + 7x^2) = -7x^2 + 1
\]

So now we have:

\[
\int \left( x + \frac{-7x^2 + 1}{x(x^2 + 7)} \right) dx
\]

### Step 3: Split the integral
We can now split the integral into two parts:

\[
\int x \, dx + \int \frac{-7x^2 + 1}{x(x^2 + 7)} \, dx
\]

1. The first integral is straightforward:

\[
\int x \, dx = \frac{x^2}{2}
\]

2. Now, for the second integral:

\[
\int \frac{-7x^2 + 1}{x(x^2 + 7)} \, dx
\]

We will decompose the integrand using partial fractions.

### Step 4: Apply partial fraction decomposition
We want to decompose:

\[
\frac{-7x^2 + 1}{x(x^2 + 7)}
\]

We assume the partial fraction decomposition to be of the form:

\[
\frac{A}{x} + \frac{Bx + C}{x^2 + 7}
\]

Multiplying both sides by \(x(x^2 + 7)\) gives:

\[
-7x^2 + 1 = A(x^2 + 7) + (Bx + C)x
\]

Expanding the right-hand side:

\[
-7x^2 + 1 = A(x^2 + 7) + Bx^2 + Cx
\]
\[
-7x^2 + 1 = (A + B)x^2 + Cx + 7A
\]

Now, equate the coefficients of \(x^2\), \(x\), and the constant term:

- For \(x^2\): \(A + B = -7\)
- For \(x\): \(C = 0\)
- For the constant term: \(7A = 1\), so \(A = \frac{1}{7}\)

Substitute \(A = \frac{1}{7}\) into \(A + B = -7\):

\[
\frac{1}{7} + B = -7
\]
\[
B = -7 - \frac{1}{7} = -\frac{50}{7}
\]

Thus, the partial fraction decomposition is:

\[
\frac{-7x^2 + 1}{x(x^2 + 7)} = \frac{1}{7x} - \frac{50x}{7(x^2 + 7)}
\]

### Step 5: Integrate each term
We now integrate the partial fractions:

1. \(\int \frac{1}{7x} \, dx = \frac{1}{7} \ln |x|\)
2. \(\int \frac{50x}{7(x^2 + 7)} \, dx\)

For the second integral, use the substitution \(u = x^2 + 7\), so \(du = 2x \, dx\), which gives:

\[
\int \frac{50x}{7(x^2 + 7)} \, dx = \frac{25}{7} \ln |x^2 + 7|
\]

### Step 6: Combine the results
Putting everything together, the integral becomes:

\[
\int \frac{x^4 + 1}{x^3 + 7x} \, dx = \frac{x^2}{2} + \frac{1}{7} \ln |x| - \frac{25}{7} \ln |x^2 + 7| + C
\]

where \(C\) is the constant of integration.

Would you like more details or have any questions on the steps?

Here are 5 questions to expand on this:

1. How do we know when to use polynomial long division in integrals?
2. Why is it necessary to factor the denominator before attempting partial fractions?
3. Can you explain why the substitution method is useful for the second integral involving \(\ln(x^2 + 7)\)?
4. How does partial fraction decomposition help simplify complex rational functions?
5. Are there any cases where partial fractions are not applicable in integration?

**Tip:** When performing polynomial long division, always subtract carefully to avoid errors that can carry over through multiple steps!

Solve the integral \(\int \frac{x^4 + 1}{x^3 + 7x} dx\) using partial fraction decomposition.

Solve the integral \(\int \frac{x^4 + 1}{x^3 + 7x} dx\) with detailed steps including polynomial long division, partial fraction decomposition, and logarithmic integration. Learn how these techniques simplify complex rational functions in integral calculus.

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