Let's solve the integrals for problems 18 and 27 from the image.

Problem 18

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

Solution:

To solve this, we can split the integral as follows: $\int (2 - 3x) \, dx = \int 2 \, dx - \int 3x \, dx$

Now, integrate each term separately:

1. For $\int 2 \, dx$: $\int 2 \, dx = 2x$

2. For $\int 3x \, dx$: $\int 3x \, dx = \frac{3x^2}{2}$

Now, combine these results: $\int (2 - 3x) \, dx = 2x - \frac{3x^2}{2} + C$ where $C$ is the constant of integration.

Final Answer for Problem 18:

$\int (2 - 3x) \, dx = 2x - \frac{3x^2}{2} + C$

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Problem 27

$\int (\sqrt{x} + 3 \sqrt{x}) \, dx$

Solution:

We can rewrite the expression as follows: $\int (\sqrt{x} + 3\sqrt{x}) \, dx = \int (4\sqrt{x}) \, dx = \int 4x^{1/2} \, dx$

Now, apply the power rule for integration: $\int 4x^{1/2} \, dx = 4 \cdot \frac{x^{1/2 + 1}}{1/2 + 1} = 4 \cdot \frac{x^{3/2}}{3/2} = \frac{4 \cdot 2}{3} x^{3/2} = \frac{8}{3} x^{3/2}$

Final Answer for Problem 27:

$\int (\sqrt{x} + 3 \sqrt{x}) \, dx = \frac{8}{3} x^{3/2} + C$ where $C$ is the constant of integration.

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Would you like further details on these solutions or have any questions?

Further Related Questions:

1. How would you handle integrals with variable substitutions?
2. What is the power rule for integration and when is it applied?
3. How do you integrate functions with fractional exponents?
4. Can you integrate functions with multiple terms separately?
5. What are the common integration techniques for polynomials?

Tip:

Always check your integration by differentiating the result. If you return to the original function, your integration is likely correct!