To solve the indefinite integral

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

we can use the substitution method. Let's proceed with the steps:

Step 1: Choose a Substitution

Notice that the denominator $4 + 3x^4$ has a derivative that includes $x^3$ as a factor, suggesting that we could try $u = 4 + 3x^4$.

1. Let $u = 4 + 3x^4$.
2. Differentiate both sides with respect to $x$: $\frac{du}{dx} = 12x^3.$
3. Solve for $dx$: $dx = \frac{du}{12x^3}.$

Step 2: Substitute into the Integral

Now, rewrite the integral in terms of $u$: $\int \frac{x^3}{4 + 3x^4} \, dx = \int \frac{x^3}{u} \cdot \frac{du}{12x^3}.$

The $x^3$ terms cancel out, simplifying to: $\int \frac{1}{u} \cdot \frac{du}{12} = \frac{1}{12} \int \frac{1}{u} \, du.$

Step 3: Integrate with Respect to $u$

The integral $\int \frac{1}{u} \, du$ is a standard form: $\frac{1}{12} \int \frac{1}{u} \, du = \frac{1}{12} \ln |u| + C.$

Step 4: Substitute Back $u = 4 + 3x^4$

Substitute $u = 4 + 3x^4$ back into the solution: $\frac{1}{12} \ln |4 + 3x^4| + C.$

Final Answer

$\int \frac{x^3}{4 + 3x^4} \, dx = \frac{1}{12} \ln |4 + 3x^4| + C.$

Would you like more details on any specific step, or do you have any questions?

Related Questions:

1. What other substitution methods are useful in solving integrals like this?
2. How can we determine if a substitution is appropriate for a given integral?
3. Are there cases where substitution fails and a different approach is needed?
4. What are the conditions for using the logarithmic integration rule in calculus?
5. How does the constant $C$ affect the solution of an indefinite integral?

Tip:

Always look for expressions in the integral that suggest a derivative relationship; it can guide you in choosing an effective substitution.