To solve this integral using substitution, let's analyze the given expression:

$\int 6x^5 \left( x^6 \right)^5 \, dx$

Step 1: Substitution

To simplify the integral, we can use substitution. Let's set: $u = x^6$ Then, taking the derivative of both sides with respect to $x$: $\frac{du}{dx} = 6x^5 \Rightarrow dx = \frac{du}{6x^5}$

This substitution means that $6x^5 \, dx = du$. Rewriting the integral in terms of $u$: $\int 6x^5 \left( x^6 \right)^5 \, dx = \int u^5 \, du$

Step 2: Integrate with respect to $u$

Now we integrate: $\int u^5 \, du = \frac{u^6}{6} + C$

Step 3: Substitute back $u = x^6$

Now, replace $u$ with $x^6$: $\frac{u^6}{6} + C = \frac{(x^6)^6}{6} + C = \frac{x^{36}}{6} + C$

So, the result of the integration is: $\int 6x^5 \left( x^6 \right)^5 \, dx = \frac{x^{36}}{6} + C$

Verifying Without Substitution

Notice that $\left( x^6 \right)^5 = x^{30}$, so the integral can be rewritten as: $\int 6x^5 \cdot x^{30} \, dx = \int 6x^{35} \, dx$

Now integrate directly: $\int 6x^{35} \, dx = 6 \cdot \frac{x^{36}}{36} + C = \frac{x^{36}}{6} + C$

Both methods give the same result: $\frac{x^{36}}{6} + C$

Would you like more details on any of the steps?

Related Questions

1. How do we determine the substitution variable for complex integrals?
2. Why is the derivative of $x^6$ equal to $6x^5$?
3. What is the importance of confirming results using different methods?
4. How does substitution simplify certain integrals?
5. Can this substitution method be used for all power integrals?

Tip

When dealing with powers inside a function, consider substitution for simplification. This often reduces complex powers and expressions into simpler ones to integrate.