Let's solve each integral step by step:

(a) $\int 15x \, dx$

The general rule for integration is: $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$ Here, $n = 1$: $\int 15x \, dx = 15 \cdot \frac{x^{1+1}}{1+1} + C = 15 \cdot \frac{x^2}{2} + C = \frac{15x^2}{2} + C$

(b) $\int (3x^2 - 4) \, dx$

Using the rule for integration term by term:

1. For $3x^2$, $n = 2$: $\int 3x^2 \, dx = 3 \cdot \frac{x^{2+1}}{2+1} = x^3$
2. For $-4$, $\int -4 \, dx = -4x$: $\int (3x^2 - 4) \, dx = x^3 - 4x + C$

(c) $\int \frac{3}{x^4} \, dx$

Rewrite $\frac{3}{x^4}$ as $3x^{-4}$ and integrate: $\int 3x^{-4} \, dx = 3 \cdot \frac{x^{-4+1}}{-4+1} = 3 \cdot \frac{x^{-3}}{-3} = -\frac{1}{x^3} + C$

(d) $\int (5x^2 + 2\cos(x)) \, dx$

Integrate term by term:

1. For $5x^2$, $n = 2$: $\int 5x^2 \, dx = 5 \cdot \frac{x^{2+1}}{2+1} = \frac{5x^3}{3}$
2. For $2\cos(x)$: $\int 2\cos(x) \, dx = 2\sin(x)$ Thus: $\int (5x^2 + 2\cos(x)) \, dx = \frac{5x^3}{3} + 2\sin(x) + C$

Final Results:

(a) $\frac{15x^2}{2} + C$
(b) $x^3 - 4x + C$
(c) $-\frac{1}{x^3} + C$
(d) $\frac{5x^3}{3} + 2\sin(x) + C$

Would you like more detailed explanations or have any questions?

Here are 5 related questions for further exploration:

1. How do we apply the power rule for definite integrals?
2. What happens if the exponent in $\int x^n \, dx$ is $-1$?
3. How can we integrate functions involving logarithms?
4. Can you explain why the integral of $\cos(x)$ is $\sin(x)$?
5. How do we solve integrals with trigonometric substitutions?

Tip: Always double-check that the power rule applies before integrating, especially for fractional or negative powers.