To solve these two integrals, let's analyze each one separately.

Integral 1: $f_1(x) = \int (2x + 1.5) \cos \left( \frac{\pi n x}{2} \right) \, dn$

This integral involves a product of a linear function in $x$ and a cosine function. We’ll treat $x$ as a constant with respect to $n$ since we’re integrating with respect to $n$.

1. Rewrite the integral: $f_1(x) = \int (2x + 1.5) \cos \left( \frac{\pi n x}{2} \right) \, dn$

Since $2x + 1.5$ is a constant factor with respect to $n$, we can factor it out:

$f_1(x) = (2x + 1.5) \int \cos \left( \frac{\pi n x}{2} \right) \, dn$

2. Now, integrate $\cos \left( \frac{\pi n x}{2} \right)$ with respect to $n$: $\int \cos \left( \frac{\pi n x}{2} \right) \, dn = \frac{2}{\pi x} \sin \left( \frac{\pi n x}{2} \right) + C$

3. Substitute back: $f_1(x) = (2x + 1.5) \cdot \frac{2}{\pi x} \sin \left( \frac{\pi n x}{2} \right) + C$

Thus, $f_1(x) = \frac{(2x + 1.5) \cdot 2}{\pi x} \sin \left( \frac{\pi n x}{2} \right) + C$

Integral 2: $f_2(x) = \int (-x + 0.5) \sin \left( \frac{\pi n x}{2} \right) \, dn$

Similarly, for this integral:

1. Rewrite the integral: $f_2(x) = \int (-x + 0.5) \sin \left( \frac{\pi n x}{2} \right) \, dn$

Factor out $-x + 0.5$:

$f_2(x) = (-x + 0.5) \int \sin \left( \frac{\pi n x}{2} \right) \, dn$

2. Integrate $\sin \left( \frac{\pi n x}{2} \right)$ with respect to $n$: $\int \sin \left( \frac{\pi n x}{2} \right) \, dn = -\frac{2}{\pi x} \cos \left( \frac{\pi n x}{2} \right) + C$

3. Substitute back: $f_2(x) = (-x + 0.5) \cdot \left(-\frac{2}{\pi x} \cos \left( \frac{\pi n x}{2} \right)\right) + C$

Simplify: $f_2(x) = \frac{(-x + 0.5) \cdot 2}{\pi x} \cos \left( \frac{\pi n x}{2} \right) + C$

Final Answers

1. $f_1(x) = \frac{(2x + 1.5) \cdot 2}{\pi x} \sin \left( \frac{\pi n x}{2} \right) + C$
2. $f_2(x) = \frac{(-x + 0.5) \cdot 2}{\pi x} \cos \left( \frac{\pi n x}{2} \right) + C$

Would you like more details on any step, or do you have further questions?

Related Questions

1. How do you integrate functions involving trigonometric terms?
2. What are typical techniques for integrals of the form $f(x) \cos(ax)$ or $f(x) \sin(ax)$?
3. How does treating variables as constants affect integration steps?
4. How would the result differ if we integrated with respect to $x$ instead?
5. How do you interpret the constant $C$ in indefinite integrals?

Tip:

When integrating trigonometric functions with a linear argument (like $\cos(kx)$ or $\sin(kx)$), remember to adjust for the coefficient of $x$ by dividing by it after integration.