Good evening! Let's go through each of these integrals step-by-step.

Direct Integration Method:

1) ∫ (3x² + 6x + 8) dx

To solve this, we integrate term by term:

• ∫ 3x² dx = (3/3) x³ = x³
• ∫ 6x dx = (6/2) x² = 3x²
• ∫ 8 dx = 8x

Thus, the indefinite integral is: $\int (3x² + 6x + 8) dx = x³ + 3x² + 8x + C$ where $C$ is the constant of integration.

2) ∫ (√x + 1)(x√x + 1) dx

First, expand the integrand: $(\sqrt{x} + 1)(x\sqrt{x} + 1) = \sqrt{x} \cdot x\sqrt{x} + \sqrt{x} \cdot 1 + 1 \cdot x\sqrt{x} + 1 \cdot 1$ $= x^{3/2} + \sqrt{x} + x^{3/2} + 1$ $= 2x^{3/2} + \sqrt{x} + 1$

Now, integrate term by term:

• ∫ 2x^{3/2} dx = $2 \times \frac{2}{5} x^{5/2} = \frac{4}{5} x^{5/2}$
• ∫ x^{1/2} dx = $\frac{2}{3} x^{3/2}$
• ∫ 1 dx = $x$

Thus, the indefinite integral is: $\int (2x^{3/2} + x^{1/2} + 1) dx = \frac{4}{5} x^{5/2} + \frac{2}{3} x^{3/2} + x + C$

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Using Substitution (Invariance of the Integration Formula):

1) ∫ (x + 9)¹³ dx

Let’s use the substitution $u = x + 9$. Therefore, $du = dx$. The integral becomes: $\int u^{13} du$ Integrating: $\frac{u^{14}}{14} = \frac{(x + 9)^{14}}{14} + C$

2) ∫ ⁵√(7 - 3x)⁴ dx

Let’s make the substitution $u = 7 - 3x$, so that $du = -3 dx$, or $dx = -\frac{1}{3} du$. The integral becomes: $\int \left( u \right)^4 \left( -\frac{1}{3} \right) du = -\frac{1}{3} \int u^4 du$ Integrating: $-\frac{1}{3} \times \frac{u^5}{5} = -\frac{1}{15} (7 - 3x)^5 + C$

3) ∫ eˣ / x² dx

Here, we use the substitution $u = x$, hence $du = dx$, and the integral becomes: $\int \frac{e^u}{u^2} du$ This integral doesn't have an elementary closed form (it’s a non-elementary integral), but it is expressed in terms of the exponential integral function, denoted as $\text{Ei}(x)$.

So, the solution is expressed as: $\int \frac{e^x}{x^2} dx = \text{Ei}(x) + C$

4) ∫ eˣ sin(eˣ) dx

Let’s use the substitution $u = e^x$, so that $du = e^x dx$, and the integral becomes: $\int \sin(u) du$ Integrating: $-\cos(u) = -\cos(e^x) + C$

5) ∫ sin⁵(x) cos(x) dx

Let’s use the substitution $u = \sin(x)$, so that $du = \cos(x) dx$. The integral becomes: $\int u^5 du$ Integrating: $\frac{u^6}{6} = \frac{\sin^6(x)}{6} + C$

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Summary of Solutions:

1. ∫ (3x² + 6x + 8) dx = $x³ + 3x² + 8x + C$
2. ∫ (√x + 1)(x√x + 1) dx = $\frac{4}{5} x^{5/2} + \frac{2}{3} x^{3/2} + x + C$
3. ∫ (x + 9)¹³ dx = $\frac{(x + 9)^{14}}{14} + C$
4. ∫ ⁵√(7 - 3x)⁴ dx = $-\frac{(7 - 3x)^5}{15} + C$
5. ∫ eˣ / x² dx = $\text{Ei}(x) + C$ (non-elementary)
6. ∫ eˣ sin(eˣ) dx = $-\cos(e^x) + C$
7. ∫ sin⁵(x) cos(x) dx = $\frac{\sin^6(x)}{6} + C$

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Would you like any further details or clarifications on any of these? Here are some related questions you might find interesting:

1. How would you approach integrals involving trigonometric functions raised to higher powers, like sin⁷(x)?
2. Can you explain the concept of non-elementary integrals and when we use special functions like the Exponential Integral?
3. What is the connection between substitution and differential equations?
4. How do you verify if an integral has a simple closed-form solution or if it requires special functions?
5. How does substitution relate to change of variables in multivariable calculus?

Tip: When you face integrals that seem difficult, try to look for substitutions that simplify the structure of the function, such as $u = e^x$, $u = x^n$, or even simple polynomial forms.