$$ \int-\text{ }\frac{6x^3}{\left(1-5x^4\right)^3} $$

We can solve the integral:

\[
I = \int -\frac{6x^3}{\left(1-5x^4\right)^3} \, dx
\]

To proceed, we'll perform a **substitution** to simplify the expression.

### Step 1: Substitution
Let:
\[
u = 1 - 5x^4
\]

Then, differentiate $ u $ with respect to $ x $:
\[
du = -20x^3 \, dx \quad \text{or} \quad dx = \frac{du}{-20x^3}
\]

### Step 2: Substituting in the integral
Now substitute in the integral:

\[
I = \int -\frac{6x^3}{u^3} \cdot \frac{du}{-20x^3}
\]

The $ x^3 $ terms cancel out, and the constants simplify:

\[
I = \int \frac{6}{20} \cdot \frac{1}{u^3} \, du
\]

Simplify the constants:

\[
I = \frac{3}{10} \int u^{-3} \, du
\]

### Step 3: Integrating
The integral of $ u^{-3} $ is:

\[
\int u^{-3} \, du = \frac{u^{-2}}{-2} = -\frac{1}{2u^2}
\]

Thus, the integral becomes:

\[
I = \frac{3}{10} \cdot \left(-\frac{1}{2u^2}\right)
\]

Simplifying:

\[
I = -\frac{3}{20u^2}
\]

### Step 4: Substitute $ u = 1 - 5x^4 $ back in:
Now, substitute $ u = 1 - 5x^4 $ back into the result:

\[
I = -\frac{3}{20(1 - 5x^4)^2} + C
\]

Thus, the final answer is:

\[
I = -\frac{3}{20(1 - 5x^4)^2} + C
\]

Would you like any further clarification or more details?

Here are 5 related questions for further exploration:
1. How do you perform integration by parts?
2. How does substitution help simplify integrals?
3. What are some common techniques for solving rational functions?
4. How can you recognize when to use trigonometric substitution?
5. What are some strategies for solving integrals involving powers of trigonometric functions?

**Tip**: When integrating complex fractions, substitution can often turn a difficult integral into something more manageable by simplifying the expression.

This page provides a detailed step-by-step solution to the integral ∫ -\frac{6x^3}{(1 - 5x^4)^3} using the substitution method. Learn how substitution simplifies complex fractions and integrals, making this problem accessible for students studying Calculus I/II. Key formulas and theorems involved include the power rule for integration and substitution techniques.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation