To solve the integral

$\int 6x^5 (1 - x^6)^5 \, dx,$

we'll use the formal substitution method. Let's proceed step by step.

Step 1: Substitution

Let $u = 1 - x^6.$ Then, $\frac{du}{dx} = -6x^5.$ So, we have $du = -6x^5 \, dx,$ or equivalently, $- du = 6x^5 \, dx.$

Now we can rewrite the integral in terms of $u$ and $du$:

$\int 6x^5 (1 - x^6)^5 \, dx = \int (u)^5 \cdot (-du).$

Simplifying, we get:

$= - \int u^5 \, du.$

Step 2: Integrate with Respect to $u$

Now, integrate $u^5$ with respect to $u$:

$- \int u^5 \, du = - \frac{u^6}{6} + C.$

Step 3: Substitute Back

Now, substitute back $u = 1 - x^6$:

$- \frac{(1 - x^6)^6}{6} + C.$

Final Answer

Thus, the indefinite integral is

$\int 6x^5 (1 - x^6)^5 \, dx = - \frac{(1 - x^6)^6}{6} + C.$

Verification by Differentiation

To verify, let's differentiate $- \frac{(1 - x^6)^6}{6} + C$ with respect to $x$:

$\frac{d}{dx} \left( - \frac{(1 - x^6)^6}{6} + C \right) = -\frac{6(1 - x^6)^5 \cdot (-6x^5)}{6}.$

Simplifying,

$= 6x^5 (1 - x^6)^5,$

which matches the original integrand. Therefore, our solution is verified.

Would you like more details or have any questions?

Here are related questions to explore:

1. How does substitution simplify the process of integration?
2. Can you apply substitution when integrating $\int x^3 (1 + x^4)^7 \, dx$?
3. How would you handle integration if the power of $(1 - x^6)$ was different?
4. What is the role of the constant of integration $C$ in indefinite integrals?
5. How does verifying by differentiation help confirm the accuracy of an integral solution?

Tip: Always verify your integration by differentiating your result; it's a useful way to ensure correctness.