https://file.aiquickdraw.com/mathbot/file/17322963460762t292ewl.jpg

from sympy import symbols, integrate, exp

# Define symbols
x = symbols('x')

# Define each integral based on the provided problems

# 1) Integral of (5e^x + 8/x)
integral_1 = integrate(5*exp(x) + 8/x, x)

# 2) Integral of (4*x**(3/2) + 8/sqrt(x))
integral_2 = integrate(4*x**(3/2) + 8*x**(-1/2), x)

# 3) Expand and integrate (3x - 4)(5x - 6)^2
expanded_3 = (3*x - 4)*(5*x - 6)**2
integral_3 = integrate(expanded_3, x)

# 4) Simplify and integrate (16x^5 - 6x^3 + 5x^2 + 2) / x^2
integral_4 = integrate((16*x**5 - 6*x**3 + 5*x**2 + 2)/x**2, x)

# 5) Definite integral of (3x^4 + 6x^3 - 2x^-3) from x = 1 to x = 4
integral_5 = integrate(3*x**4 + 6*x**3 - 2*x**(-3), (x, 1, 4))

integral_1, integral_2, integral_3, integral_4, integral_5(5*exp(x) + 8*log(x),
 16.0*x**0.5 + 1.6*x**2.5,
 75*x**4/4 - 280*x**3/3 + 174*x**2 - 144*x,
 4*x**4 - 3*x**2 + 5*x - 2/x,
 79629/80)(5*exp(x) + 8*log(x),
 16.0*x**0.5 + 1.6*x**2.5,
 75*x**4/4 - 280*x**3/3 + 174*x**2 - 144*x,
 4*x**4 - 3*x**2 + 5*x - 2/x,
 79629/80)successHere are the solutions to the integrals provided:

1. \(\int \left( 5e^x + \frac{8}{x} \right) dx = 5e^x + 8\ln(x) + C\)

2. \(\int \left( 4x^{3/2} + \frac{8}{\sqrt{x}} \right) dx = 16\sqrt{x} + 1.6x^{5/2} + C\)

3. \(\int (3x - 4)(5x - 6)^2 dx = \frac{75x^4}{4} - \frac{280x^3}{3} + 174x^2 - 144x + C\)

4. \(\int \frac{16x^5 - 6x^3 + 5x^2 + 2}{x^2} dx = 4x^4 - 3x^2 + 5x - \frac{2}{x} + C\)

5. \(\int_{1}^{4} \left( 3x^4 + 6x^3 - 2x^{-3} \right) dx = \frac{79629}{80}\)

Let me know if you need a detailed explanation for any specific step!  

**Questions to expand your understanding:**

1. How would the approach change for definite vs. indefinite integrals?
2. What methods can you use to simplify the expression before integrating?
3. Why do logarithmic terms appear in some integrals?
4. What role does substitution play in solving integrals with roots or fractions?
5. How does the presence of higher-order terms affect integration techniques?

**Tip:** Always simplify and rearrange complex expressions before attempting integration—it often reduces the chance of errors.

Find the indefinite integral of each of the following:
1) ∫ (5e^x + 8/x) dx
2) ∫ (4x^(3/2) + 8/sqrt(x)) dx
3) ∫ (3x - 4)(5x - 6)^2 dx
4) ∫ [(16x^5 - 6x^3 + 5x^2 + 2) / x^2] dx
5) ∫_1^4 (3x^4 + 6x^3 - 2x^(-3)) dx

Solve a variety of integration problems involving exponential, polynomial, and rational functions. Learn how to approach both definite and indefinite integrals with detailed step-by-step explanations. Suitable for students in Grades 11-12 and early college.

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