It seems you're asking for the indefinite integral of the following expression:

$\int \frac{4x^2 - 3x + 4}{x^3 + 4x} \, dx$

Let's solve this step by step.

Step 1: Simplify the Expression

The integral is:

$\int \frac{4x^2 - 3x + 4}{x^3 + 4x} \, dx$

First, factor the denominator:

$x^3 + 4x = x(x^2 + 4)$

Thus, the integral becomes:

$\int \frac{4x^2 - 3x + 4}{x(x^2 + 4)} \, dx$

Step 2: Break into Partial Fractions

Now, let's split the expression into simpler fractions. Write it as:

$\frac{4x^2 - 3x + 4}{x(x^2 + 4)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 4}$

We need to find constants $A$, $B$, and $C$. Multiply both sides by $x(x^2 + 4)$ to get:

$4x^2 - 3x + 4 = A(x^2 + 4) + (Bx + C)x$

Simplify:

$4x^2 - 3x + 4 = A(x^2 + 4) + Bx^2 + Cx$

$4x^2 - 3x + 4 = (A + B)x^2 + Cx + 4A$

Step 3: Equate Coefficients

Now compare the coefficients of $x^2$, $x$, and the constant terms:

• $x^2$ term: $A + B = 4$
• $x$ term: $C = -3$
• Constant term: $4A = 4$, so $A = 1$

From $A + B = 4$, and $A = 1$, we get $B = 3$.

Thus, we have:

$A = 1, \, B = 3, \, C = -3$

Step 4: Rewrite the Expression

Now rewrite the integral as:

$\int \left( \frac{1}{x} + \frac{3x - 3}{x^2 + 4} \right) dx$

Step 5: Integrate

Now, integrate each term:

1. $\int \frac{1}{x} \, dx = \ln |x|$
2. $\int \frac{3x}{x^2 + 4} \, dx$:
   • Use substitution: Let $u = x^2 + 4$, so $du = 2x \, dx$, and the integral becomes: $\frac{3}{2} \int \frac{du}{u} = \frac{3}{2} \ln |u| = \frac{3}{2} \ln (x^2 + 4)$
3. $\int \frac{-3}{x^2 + 4} \, dx$:
   • This is a standard arctangent form: $\frac{-3}{2} \arctan \left( \frac{x}{2} \right)$

Step 6: Combine the Results

Thus, the final result is:

$\ln |x| + \frac{3}{2} \ln (x^2 + 4) - \frac{3}{2} \arctan \left( \frac{x}{2} \right) + C$

Where $C$ is the constant of integration.

---

Would you like more details on any step or another explanation?

5 Related Questions

1. What is the method of partial fractions?
2. How does substitution work in integration?
3. Can you explain the arctangent integral in more detail?
4. What are standard integral forms to memorize?
5. How do you solve integrals involving logarithms?

Tip

When tackling complex integrals, always look for ways to factor the denominator or simplify the integrand before choosing a method of integration.