Question

Finding the integral of eˣ * cos(x).

a) (1/2)eˣ(sin(x) + cos(x)) + C
b) eˣ(sin(x) + cos(x)) + C
c) (1/2)eˣ(sin(x) - cos(x)) + C
d) eˣ(sin(x) - cos(x)) + C

Answer 1

The integral of eˣ * cos(x) is (1/2)eˣ(sin(x) + cos(x)) + C (Option c).

Step-by-step explanation:

To find the integral of eˣ * cos(x), we can use integration by parts. Let u = eˣ and dv = cos(x) dx. Then, du = eˣ dx and v = ∫cos(x) dx = sin(x). Applying the integration by parts formula ∫u dv = uv - ∫v du, we get:

∫eˣ * cos(x) dx = eˣ * sin(x) - ∫sin(x) * eˣ dx

Now, we can apply integration by parts again to evaluate the remaining integral. Let u = sin(x) and dv = eˣ dx. Then, du = cos(x) dx and v = ∫eˣ dx = eˣ. Applying the formula, we have:

∫sin(x) * eˣ dx = eˣ * sin(x) - ∫cos(x) * eˣ dx

Substitute this back into the original expression:

∫eˣ * cos(x) dx = eˣ * sin(x) - (eˣ * sin(x) - ∫cos(x) * eˣ dx)

Combine like terms:

∫eˣ * cos(x) dx = eˣ * sin(x) - eˣ * sin(x) + ∫cos(x) * eˣ dx

Simplify:

∫eˣ * cos(x) dx = ∫cos(x) * eˣ dx

Now, the remaining integral is the same as the original integral, but with the order of functions reversed. So, we can rewrite it as:

∫eˣ * cos(x) dx = (1/2)eˣ(sin(x) + cos(x)) + C

Therefore, the final answer is (1/2)eˣ(sin(x) + cos(x)) + C (Option c).