The integral in the image is:

$\int 2x^3 (2x^3 - 6)^6 \, dx$

Step-by-Step Solution:

Step 1: Substitution

Let: $u = 2x^3 - 6$

Differentiate $u$: $\frac{du}{dx} = 6x^2 \quad \text{or} \quad du = 6x^2 \, dx$

Rewrite $2x^3$ as $x^2 \cdot 2x$: To match ( du = 2Continuing:

We rewrite the integral with substitution: $\int 2x^3 (2x^3 - 6)^6 \, dx$

Using $u = 2x^3 - 6$, we differentiate to find: $\frac{du}{dx} = 6x^2 \quad \Rightarrow \quad dx = \frac{du}{6x^2}.$

Now, we rewrite $2x^3$ as $x^2 \cdot 2x$. Substituting back into the integral, and replacing $dx$: $\int 2x^3 (2x^3 - 6)^6 \, dx = \int u^6 \cdot \frac{du}{3}.$

Simplify: $\int u^6 \cdot \frac{du}{3} = \frac{1}{3} \int u^6 \, du.$

Step 2: Solve the Simplified Integral

The integral of $u^6$ is: $\int u^6 \, du = \frac{u^7}{7}.$

Thus: $\frac{1}{3} \int u^6 \, du = \frac{1}{3} \cdot \frac{u^7}{7} = \frac{u^7}{21}.$

Step 3: Back-Substitute $u$

Recall that $u = 2x^3 - 6$. Substituting back: $\frac{u^7}{21} = \frac{(2x^3 - 6)^7}{21}.$

Final Answer:

$\int 2x^3 (2x^3 - 6)^6 \, dx = \frac{(2x^3 - 6)^7}{21} + C,$ where $C$ is the constant of integration.

Would you like further details or explanations?

Related Questions:

1. How do we decide on substitution in an integral problem?
2. What are the properties of definite and indefinite integrals?
3. How can we apply the chain rule in reverse to verify the answer?
4. What techniques are useful when integrating polynomials raised to a power?
5. What are other examples of integrals requiring substitution?

Tip:

When tackling integrals with powers and compositions, substitution simplifies the process by reducing the problem into a basic polynomial integral!