The problem asks to evaluate the indefinite integral:

$\int (5x^2 + 1)(5x^3 + 3x - 8)^6 \, dx$

This is a composite function where one part, $(5x^3 + 3x - 8)$, is raised to a power and multiplied by another function, $5x^2 + 1$.

To solve this, we'll use the method of substitution, which is suggested by the hint to use methods from examples. Here’s a step-by-step breakdown of the solution.

Step 1: Identify the substitution

Let: $u = 5x^3 + 3x - 8$

Now, differentiate $u$ with respect to $x$ to find $\frac{du}{dx}$: $\frac{du}{dx} = 15x^2 + 3$

Thus, we have: $du = (15x^2 + 3) \, dx$

Notice that the expression $15x^2 + 3$ is a multiple of $5x^2 + 1$. In fact: $15x^2 + 3 = 3(5x^2 + 1)$ Therefore, we can rewrite $du$ as: $du = 3(5x^2 + 1) \, dx$ This implies: $\frac{du}{3} = (5x^2 + 1) \, dx$

Step 2: Substitute into the integral

Now, substitute $u$ and $\frac{du}{3}$ into the integral: $\int (5x^2 + 1)(5x^3 + 3x - 8)^6 \, dx = \int (5x^2 + 1) \cdot u^6 \, dx$ Substituting the differential $dx$ as well: $\int u^6 \cdot \frac{du}{3}$

Step 3: Integrate with respect to $u$

Now, integrate the simpler expression: $\frac{1}{3} \int u^6 \, du$

The integral of $u^6$ is: $\frac{u^7}{7}$ So the solution to the integral is: $\frac{1}{3} \cdot \frac{u^7}{7} = \frac{u^7}{21}$

Step 4: Substitute back the expression for $u$

Now, substitute back $u = 5x^3 + 3x - 8$: $\frac{(5x^3 + 3x - 8)^7}{21}$

Finally, don't forget the constant of integration: $\frac{(5x^3 + 3x - 8)^7}{21} + C$

Final Answer

The evaluated indefinite integral is: $\frac{(5x^3 + 3x - 8)^7}{21} + C$

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Would you like more details on this process? Here are five related questions to help you further understand the concept:

1. How can substitution be used to solve more complex integrals?
2. What is the importance of choosing the right substitution in integration?
3. How does the chain rule relate to the method of substitution?
4. Can you evaluate the definite integral of this function over a specific interval?
5. How would you approach this integral if the exponent on the binomial was different, say $(5x^3 + 3x - 8)^5$?

Tip: When performing substitution, always check if the differential $du$ can be expressed in terms of the original differential $dx$. If not, you may need to adjust your substitution or solve for an additional factor.