The given integral appears to be:

$\int_1^3 (4x - 6x^2)(x^3 - x^2 + 8)^9 \, dx$

Approach:

We'll use substitution to solve the integral. Let us break it down:

1. Let $u = x^3 - x^2 + 8$.
2. Then, $\frac{du}{dx} = 3x^2 - 2x$, which implies $du = (3x^2 - 2x)dx$.

Now, rewrite $4x - 6x^2$ in terms of $3x^2 - 2x$: $4x - 6x^2 = -2(3x^2 - 2x).$

Thus, the integral becomes: $\int_1^3 (4x - 6x^2)(x^3 - x^2 + 8)^9 dx = -2 \int u^9 \, du.$

Compute the integral:

The integral of $u^9$ is: $\int u^9 \, du = \frac{u^{10}}{10}.$

Substitute this back: $-2 \int u^9 \, du = -2 \cdot \frac{u^{10}}{10} = -\frac{u^{10}}{5}.$

Back-substitute $u = x^3 - x^2 + 8$:

$-\frac{u^{10}}{5} = -\frac{(x^3 - x^2 + 8)^{10}}{5}.$

Apply the limits $x = 1$ to $x = 3$:

Now evaluate: $\int_1^3 (4x - 6x^2)(x^3 - x^2 + 8)^9 dx = -\frac{(x^3 - x^2 + 8)^{10}}{5} \bigg|_1^3.$

1. When $x = 3$: $u = 3^3 - 3^2 + 8 = 27 - 9 + 8 = 26.$

2. When $x = 1$: $u = 1^3 - 1^2 + 8 = 1 - 1 + 8 = 8.$

Thus: $\int_1^3 (4x - 6x^2)(x^3 - x^2 + 8)^9 dx = -\frac{26^{10}}{5} + \frac{8^{10}}{5}.$

Final Answer:

$\boxed{-\frac{26^{10}}{5} + \frac{8^{10}}{5}}$

Would you like me to simplify further, expand the steps, or compute the numerical value?

5 Related Questions:

1. What are the steps to determine a substitution variable in integrals?
2. How can substitution be applied when there are powers of a binomial function?
3. Why is substitution effective in simplifying integrals with powers of a polynomial?
4. How do definite integrals differ from indefinite integrals in computation?
5. What are the best ways to verify substitution-based integration?

Tip:

Always check if the derivative of the substitution matches the differential in the integral for a seamless transformation.