Let's work through these three integrals step by step.

Problem 1:

$\int \frac{30x - 15}{(3x^2 - 3x + 8)^4} \, dx$

To solve this, we will use substitution. Let:

$u = 3x^2 - 3x + 8$

Then, the derivative of $u$ with respect to $x$ is:

$\frac{du}{dx} = 6x - 3$

This closely resembles the numerator, $30x - 15$, which can be factored as:

$30x - 15 = 5(6x - 3)$

Thus, we can substitute and simplify the integral. After substitution, this will be a standard power function integral.

Problem 2:

$\int_0^{\sqrt{6}} \frac{3x}{(x^2 + 2)^3} \, dx$

For this integral, we use substitution as well. Let:

$u = x^2 + 2$

The derivative of $u$ with respect to $x$ is:

$\frac{du}{dx} = 2x$

This matches the numerator $3x$ after adjusting for a constant factor. The limits of integration will also change according to the new variable $u$. After substitution, this will be a standard integral to solve.

Problem 3:

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

Here, we can apply substitution directly. Let:

$u = 3 - 2x$

The derivative of $u$ with respect to $x$ is:

$\frac{du}{dx} = -2$

This simplifies the integral into a polynomial form which can then be expanded and integrated term by term.

Would you like detailed steps and solutions for each of these integrals, or do you have any specific questions about any of them?

Five Related Questions:

1. How do you decide when to use substitution in integrals?
2. What is the process for changing the limits of integration during substitution?
3. How can you identify which substitution will simplify a given integral?
4. How do polynomial expansions affect the process of integration?
5. What are common mistakes to avoid during integration by substitution?

Tip:

Always verify the derivative of your substitution choice to ensure it matches the structure of the integrand.