Let me analyze the problems in your uploaded image and solve them step by step:

Problem 1:

Part (a):

We are asked to evaluate the integral: $\int \big(8\cos(x) + \sec^2(x)\big) \, dx$

• For $\int 8\cos(x) \, dx$, we have: $\int 8\cos(x) \, dx = 8\sin(x)$

• For $\int \sec^2(x) \, dx$, we use the standard result: $\int \sec^2(x) \, dx = \tan(x)$

Adding these, we get: $\int \big(8\cos(x) + \sec^2(x)\big) \, dx = 8\sin(x) + \tan(x) + C$

Part (b):

We are asked to evaluate the integral: $\int \left(x^6 + \frac{3\sqrt{x}}{x^4}\right) \, dx$

Simplify the second term: $\frac{3\sqrt{x}}{x^4} = 3x^{-3.5}$

Thus, the integral becomes: $\int \big(x^6 + 3x^{-3.5}\big) \, dx$

• For $\int x^6 \, dx$, use the power rule: $\int x^6 \, dx = \frac{x^7}{7}$

• For $\int 3x^{-3.5} \, dx$, use the power rule: $\int 3x^{-3.5} \, dx = 3 \cdot \frac{x^{-2.5}}{-2.5} = -\frac{6}{5}x^{-2.5}$

Combining these: $\int \left(x^6 + \frac{3\sqrt{x}}{x^4}\right) \, dx = \frac{x^7}{7} - \frac{6}{5}x^{-2.5} + C$

Problem 2:

We are asked to evaluate: $\int \cos(7x) \, dx$

Let $u = 7x$, so $du = 7 \, dx$ or $dx = \frac{du}{7}$. Substitute into the integral: $\int \cos(7x) \, dx = \frac{1}{7} \int \cos(u) \, du$

The integral of $\cos(u)$ is $\sin(u)$, so: $\frac{1}{7} \int \cos(u) \, du = \frac{1}{7} \sin(u)$

Substitute back $u = 7x$: $\int \cos(7x) \, dx = \frac{1}{7} \sin(7x) + C$

Problem 3:

We are tasked to find $f(x)$ such that: $f'(x) = x^2 - 6x \quad \text{and} \quad f(1) = 2$

Integrate $f'(x)$: $f(x) = \int (x^2 - 6x) \, dx$

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

• For $\int -6x \, dx$: $\int -6x \, dx = -3x^2$

Combine: $f(x) = \frac{x^3}{3} - 3x^2 + C$

To find $C$, use the condition $f(1) = 2$: $f(1) = \frac{1^3}{3} - 3(1^2) + C = 2$

Simplify: $\frac{1}{3} - 3 + C = 2$

$C = 2 + 3 - \frac{1}{3} = 5 - \frac{1}{3} = \frac{15}{3} - \frac{1}{3} = \frac{14}{3}$

Thus: $f(x) = \frac{x^3}{3} - 3x^2 + \frac{14}{3}$

If you'd like further clarification or detailed steps, let me know! Here are 5 related questions to explore further:

1. How do you verify the correctness of an antiderivative?
2. What are the rules for integrating functions involving trigonometric terms?
3. How do substitution methods simplify certain integrals?
4. What is the significance of initial conditions in determining constants of integration?
5. How do power rules change for negative or fractional exponents?

Tip: Always double-check your work by differentiating your final answer to ensure it matches the original integrand!