Question

Determine the following particular integrals:
1.1 1/D^2 +4 {2 sin x cos x + 3 cos x}

Answer 1

the value of the given particular integral is 0 because 0 + 0 = 0.

Explanation:

We are given the following integral:

1/((D^2) +4){2 sin(x) cos(x) + 3 cos(x)}

Let's simplify the denominator first:

(D^2 + 4) = (D^2 + 2^2)

This can be written as:

(D + 2i)(D - 2i)

Now let's express the numerator in partial fractions:

2 sin(x) cos(x) + 3 cos(x) = A(D + 2i) + B(D - 2i)

Solving for A and B:

Let D = -2i, then we have:

A(-2i + 2i) = 3(-2i)

0 = -6i

This implies that A = 0.

Similarly, when we let D = 2i, we obtain:

B(2i - 2i) = 3(2i)

0 = 6i

This implies that B = 0.

Therefore, the original integral simplifies to:

0 + 0 = 0